String Principal Bundles and Courant Algebroids

Home , Atiyah algebroid, Courant bracket, Lie algebroid, String group

String principal bundles and Courant algebroids

Yunhe Sheng, Xiaomeng Xu and Chenchang Zhu

Abstract Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids encode the infinitesimal symmetries of S1-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the noncommutative analogue of exact Courant algebroids. In this article, we explore what the “principal bundles” behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connection data. We prove a lifting theorem for the stack of string principal bundles with connections and show the multiplicity of the lifts once they exist. This is a differential geometrical refinement of what is known for string structures by Redden, Waldorf and Stolz-Teichner. We also extend the result of Bressler and Chen-Stiénon-Xu on extension obstruction involving transitive Courant algebroids to the case of transitive Courant algebroids with connections, as a lifting theorem with the description of multiplicity once liftings exist. At the end, we build a morphism between these two stacks. The morphism turns out to be neither injective nor surjective in general, which shows that the process of associating the “higher Atiyah algebroid” loses some information and at the same time, only some special transitive Courant algebroids come from string bundles.

Contents

1 Introduction 2

2 Preliminaries on prestacks, stacks and the plus construction 4

3 (3, 1)-sheaf of string principal bundles 6 3.1 Finite dimensional model of Stringp(G) ...... 6 p+ 3.2 (3, 1)-sheaf BString(n)c of string principal bundles with connection data ...... 8 3.3 Lifting Theorem and Comparison ...... 12

p+ 4 (2, 1)-sheaf TCc of transitive Courant algebroids with connections 17

5 Morphism from the string sheaf to the transitive Courant sheaf 24 p+ p+ 5.1 Construction of the morphism Φ: BString(n)c → TCc ...... 25 p+ p+ 5.2 Property of the morphism Φ: BString(n)c → TCc ...... 29

A Appendix 31 p+ A.1 (2, 1)-sheaf TLc of transitive Lie algebroids with connections ...... 31 p+ A.2 (2, 1)-sheaf ECc of exact Courant algebroids with connections ...... 33 A.3 Gluing transitive Courant algebroids via local data ...... 35 A.4 Inner automorphisms of transitive Courant algebroids ...... 37

0Keyword:Courant algebroids, string principal bundles, Pontryagin classes 0MSC: 18B40, 18D35, 46M20, 53D17, 58H05.

1 1 Introduction

Just like Atiyah Lie algebroids encode the infinitesimal symmetries of G-principal bundles [3, 31], exact Courant algebroids are believed to encode the infinitesimal symmetries of U(1)-gerbes (or equivalently BU(1)-2-principal bundles) [10, 21, 35, 42]. Then it is proven later in [18] that the infinitesimal symmetries of an S1-gerbe equipped with a connective structure are precisely encoded by an exact Courant algebroid with a connection. Transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative analogue of exact Courant algebroids. In this article, we explore what the “principal bundles” behind transitive Courant algebroids are. First, we notice that there are topological obstructions for the existence of such transitive Courant algebroids. In [9], Bressler discovered that the obstruction to extend an Atiyah Lie algebroid to a transitive Courant algebroid is given by the real first Pontryagin class. This is further fully generalized to the case of any regular Courant algebroid by Chen-Stiénon-Xu [17] in a differential geometry setting, where the authors gave a complete classification result. In fact, Ševera outlined some very nice ideas to classify transitive Courant algebroids in a series of private letter exchanges with Weinstein [42]. The role of the Pontryagin class is further developed in his later works [43, 44]. We then notice that the first Pontryagin class arises as an obstruction in another domain. Motivated by Stolz-Teichner’s program of topological modular forms [48], Redden [34] defined a string class on a Spin(n)-principal bundle P¯ → M, as a class ξ ∈ H3(P,¯ Z), such that for every point p ∈ P¯ the associated ¯ 3 inclusion ip : Spin(n) → P by g 7→ g · p pulls back ξ to the standard generator of H (Spin(n), Z). He further proved that the obstruction for a Spin(n)-principal bundle P¯ over M to admit string classes is 1 ¯ 4 provided by the integer class of half the first Pontryagin class 2 p1(P ) ∈ H (M, Z). In [50], Waldorf proved that such string classes on P¯ are in one-to-one correspondence with the isomorphism classes of trivializations of the Chern-Simons 2-gerbe over P¯, where the Chern-Simons 2-gerbe is again characterised 1 ¯ by 2 p1(P ). Topologically, Stolz and Teichner view the above string structure on a Spin(n)-principal bundle ¯ M −→P BSpin(n) as a lift of the structure group of P¯ from Spin(n) to a certain three-connected extension, the string group String(n), BString(n) :

P¯ M / BSpin(n).

1 Thus, if we realise “String(n)-principal bundles” in diﬀerential geometry, one may interpret 2 p1 as the lifting obstruction of a Spin(n)-principal bundle to a String(n)-principal bundle. A more familiar fact in this style is that the lifting obstruction of a SO(n)-principal bundle M −→P BSO(n) to a Spin(n)-principal P˜ bundle M −→ BSpin(n) is given by w2(P ). In fact, there is a whole sequence, called the Whitehead tower:

· · · → BString(n) → BSpin(n) → BSO(n) → BO(n),

1 with obstruction w1, w2 and 2 p1 respectively. Here w1 and w2 are the first and second Stiefel-Whitney classes. Since the topological obstruction for both transitive Courant algebroids and String(n)-principal bundles is provided by the first Pontryagin class, we naturally believe that the principal bundle behind a transitive Courant algebroid is exactly a String(n)-principal bundle. This belief is also supported by another observation from T-duality. Let us start with two T-dual torus bundles X, Xˆ over M, and matching T-dual S1-gerbes G → X and Gˆ → Xˆ, Bouwknegt-Evslin-Mathai [7] and Bunke-Schick [12] proved that the twisted K-theory for the T-dual pairs are isomorphic, that is, there is an isomorphism between twisted K-groups K•(X, G) =∼ K•(X,ˆ Gˆ). On the level of differential geometric objects, Cavalcanti-Gualtieri [16] proved that the exact Courant algebroid associated to the T- dual S1-gerbes are the same. Now we extend this story Spin(n)-equivariantly. We begin with two T-dual torus bundles X, Xˆ over M, and their matching T-dual string structures (P, ξ) → X and (P,ˆ ξˆ) → Xˆ,

2 where P and Pˆ are Spin(n)-principal bundles over X and Xˆ respectively, and ξ, ξˆ are string classes on P and Pˆ respectively. Leaving alone what the cohomological invariants should be, Baraglia and Hekmati [5] showed that, on the level of differential geometric objects, the transitive Courant algebroids associated to both sides are isomorphic. In this article, we realise String(n)-principal bundles and their connections as differential geometric p+ objects by describing the entire (3, 1)-sheaf (or 2-stack) BString(n)c . Then we make the connection between transitive Courant algebroids and string principal bundles explicit and functorial by constructing a morphism between their corresponding stacks. For this purpose, first we study what a String(n)-principal bundle with connection data really is. As we have seen, String(n) is a three-connected cover of Spin(n), and this forces the model of String(n) to be either infinite-dimensional or finite-dimensional however higher (namely being a Lie 2-group)1. We take the second approach with the model of Schommer-Pries [39] for String(n). The advantage of this model is that the spaces it involves are all nice finite dimensional manifolds, thus there is no additional analytic difficulty when solving equations or constructing covers; at the same time, this is paid off by algebraic difficulty of chasing through various pages of spectral sequences of cohomological calculation. p First we construct a (3, 1)-presheaf of String(n)-principal bundles with connection data BString(n)c p+ and complete it into a (3, 1)-sheaf (or a 2-stack) BString(n)c using the plus construction. This is essentially to build a String(n)-principal bundle with a connection from local data and gluing conditions in the fashion of Breen-Messing. Breen and Messing studied connections for gerbes in their original work [8]. See [15] for more details. We also notice that in a recent work [51], connections for 2-principal bundles of strict 2-groups are studied both locally and globally. However, the finite-dimensional differential geometric model for String(n) is a non-strict Lie 2-group. This forces us to develop our own formula instead of using existing results in literature. We also noticed that [22] develops a very nice model which explains connection data for all L∞-algebras based on the infinite dimensional integration model of L∞- algebras in [26, 28]. However, to relate their infinite dimensional model to our formula of describing connection with descent equations (Eqs. (5)-(8)), one needs to solve variables on ∆n in their model in some Lie type of PDE’s. This is a non-trivial task and we delay it to a future project with more specific focus. To justify our construction, we prove directly the lifting theorem that one expects for String(n)- principal bundles and provide a comparison to previous string concepts of Stolz-Teichner, Redden and Waldorf respectively in Section 3.3:

P¯ Theorem 1.1. Given a Spin(n)-principal bundle with a connection M −→ BSpin(n)c,

Pc p+ (i) it lifts to an object in M −→ BString(n)c ,

p+ BString(n)c (1) 9

P¯ M / BSpin(n)c,

1 ¯ if and only if 2 p1(P ) = 0; (ii) if a lift in (1) exists, then the isomorphism classes of diﬀerent lifts form a torsor of the Deligne d log cohomology group H2(M,U(1) −−−→ Ω1 −→d Ω2) mod out by a certain subspace I.

p+ Why will our construction BString(n)c make this theorem work? This is answered by the most technical part of the proof, that is to calculate a certain Čech cocycle coming from the associator in the p+ 1 model of BString(n)c really represents 2 p1. This is done in Lemma 3.5, where we simply chased through several layers (explained in Section 3.3) all the multiplication bibundles and their 2-morphisms in [39]. It is like to go through a tic-tac-toe process in a 2-group.

1There are also models which are both higher and inﬁnite-dimensional.

3 After this, we build the (2, 1)-sheaf (or 1-stack) of transitive Courant algebroids with connections. We benefit much from [17] where transitive Courant algebroids and their gauge transformations are well studied. However, the gauge transformations which preserve the connection data are still needed to be specified. We thus have additional equations in the definition of 1-morphisms (see Eqs. (33)-(35)). To make the construction mathematically strict, however at the same time avoiding the routine checking of p gluing conditions of stacks over several layers, as before, we first construct a (2, 1)-presheaf TCc by simply mapping to the category of standard transitive Courant algebroids with connections and their gauge p+ transformations. Then we complete it to a (2, 1)-sheaf TCc using Nikolaus-Schweigert’s plus construction [32]. We prove that the gluing result gives us exactly a transitive Courant algebroid (not necessarily standard) with a connection and its gauge transformations. This in turn justifies our construction of the transitive Courant algebroid stack. There is a subtle difference between our construction of the transitive Courant algebroid stack and the one in [9]. In [9], the stack is directly taken to be a functor mapping to the category of transitive Courant algebroids (not just standard ones), however the checking of gluing conditions seems to be omitted. Also in the language of stacks, a recent work [33] has studied the relation between twisted Courant algebroids and shifted symplectic Lie algebroids, and has further hinted an even higher correspondence of our type involving fivebrane structures. In the end, we construct a morphism from the (3, 1)-sheaf of String(n)-principal bundles with connections to the (2, 1)-sheaf of transitive Courant algebroids with connections for Spin(n). To achieve this, we only need to build a morphism on the presheaf level since the plus construction is functorial. It turns out that the difficulty of the construction lies on the level of morphisms, that is, to construct the gauge transformation of transitive Courant algebroids associated to that of String(n)-principal bundles. The formula of the symmetric part of the (3, 1)-position in the gauge transformation remains rather mysterious. Šev- era suggests us some connection to Alekseev-Malkin-Meinrenken’s theory on group valued moment maps [2]. We reserve it for future investigation. We remark in Appendix A.4 that these gauge transformations are all inner ones noticed by Ševera. Similar results of these inner automorphisms are also studied in [24] in another setting. We further verify that this morphism from the string stack to the Courant stack is neither injective nor surjective. This tells us that the process of associating a “higher Atiyah algebroid” to a String principal 2-bundle loses some information, and, at the same time, not all transitive Courant algebroids come from this process. Recently, partially inspired by our work, the notion of a holomorphic string algebroid was introduced in [25], and showed to be the Atiyah algebroid of holomorphic principal bundles for the complexified string group. In a certain sense, [5] already gives transitive Courant algebroid with input a String structure. Never- theless, since only differential forms however not Čech cocycle appeared in [5], the topology aspect is not there. To grasp the integer string class h in their construction, which is a lift of the curvature 3-form H on M of a String principal 2-bundle P → M to the underlying Spin(n) principal bundle P¯, is not an easy task. The first time this is grasped, is through a geometric construction by Waldorf in [50] realising the string class as DD class of a certain gerbe. For us, we descend the Chern-Simon 3-form, whose lift differs from h by an exact form, to an integer Čech class, which comes from the associator of String(n). From this point of view, one of the merits of our paper is that we descend the Chern-Simon 3-form until the Čech level and the topological aspect is clearly grasped. In a recent work [1], these descent equations are further studied in a universal setting and proved to be closely related to Kashiwara-Vergne theory and Drinfeld associators for the last step of descending to the Čech level. Of course, this topological aspect will not be seen by Courant algebroid since latter is a differential geometric gadget. However, if one wants to know a more precise picture, for example, whether all transitive Courant algebroids are hit by String principal 2-bundles (surjectivity of our functor Φ), or whether different String principal 2-bundles may correspond to the same Courant algebroid (injectivity of Φ), then it is helpful to have information from the topological aspect.

4 2 Preliminaries on prestacks, stacks and the plus construction

Recall that an (n + 1, 1)-presheaf over a category M is a (higher) functor Mop → nGpd to the higher category of n-groupoids, where n ∈ {0, 1, 2,... } ∪ {∞}. Here 0-groupoids are interpreted as sets. There- fore, a (1, 1)-presheaf (or a presheaf) over a category M is a functor Mop → Sets to the category of sets; a (2, 1)-presheaf over a category M is a (higher) functor Mop → Gpd to the 2-category of (1-)groupoids; and a (3, 1)-presheaf over a category M is a (higher) functor Mop → 2Gpd to the higher category of 2-groupoids. These are all the cases that we will use in this paper. Then we perform a plus construction (namely a procedure of higher sheafification) to obtain the corresponding sheaves. Sometimes, (2, 1)- sheaves are also called stacks, and (3, 1)-sheaves are called 2-stacks. The model we use is as in [32, Section 2]. For technical details, we refer readers to this paper and the references therein. Here we briefly recall the model we use for 2-groupoids. Our model for a 2-groupoid (in the sense of Duskin and Glenn [20, 27]) is a simplicial set satisfying Kan conditions Kan(n, j) for all n ≥ 1 and 0 ≤ j ≤ n and strict Kan conditions Kan!(n, j) for all n ≥ 3 and 0 ≤ j ≤ n. Readers who are not familiar with Kan conditions may equivalently understand it as a bicategory [6], whose 2-morphisms are invertible and whose 1-morphisms are all invertible up to 2-morphisms. The compositions of 1-morphisms are associative up to an associator, and the associator in turn satisfies a higher coherence condition. For the precise definition, we refer to [47, Definition 5.2], where a semi-strict Lie 2-groupoid is defined. If we equip the object therein with discrete topology, we obtain what a 2-groupoid is. Let us also recall the equivalence between the two different descriptions: if we start with a simplicial set X• satisfying the above Kan condition, then we take C0 = X0 on the object level; C1 = X1 on the 1-morphism −1 level; C2 = d0 (s0(X0)) on the 2-morphism level, we obtain a bicategory (C0,C1,C2) satisfying required conditions; as for the other direction, we take X0 = C0, X1 = C1 and X2 = C1 ×C0 C1. For details we refer to [55, Section 4]. Now we shortly recall the process of the plus construction in the case when M = Mfd is the category of differential manifolds for our application. Given a (3, 1)-presheaf F : Mfdop → 2Gpd, the plus construction in [32] gives us a (3, 1)-sheaf F + : Mfdop → 2Gpd. To describe this (3, 1)-sheaf, we first need to take the homotopy limit holim F(U(M)•) for an open cover {Ui} of M over the Čech simplicial manifold

∂0 ∂0,∂1 U(M)• = tUi ks tUij jt tUijk .... (2) ∂1 ∂2

Let us describe holim F(U(M)•) explicitly: the result is a 2-groupoid.

• Its object consists of

Ob0 an element θ = (θi) ∈ F(tUi)0;

gij Ob1 an element g = (gij) ∈ F(tUij)1, which is a 1-morphism θi|Uij ←−− θj|Uij , or equivalently, a ∗ g ∗ 1-morphism ∂1 θ ←− ∂0 θ;

Ob2 an element a = (aijk) ∈ F(tUijk)2, which is a 2-morphism a : gij ◦ gjk ⇐ gik; ∗ ∗ ∗ ∗ Ob3 pentagon condition for a, that is, (id ◦h ∂0 a) ◦ ∂2 a = α(gij, gjk, gkl) ◦ (∂3 a ◦h id) ◦ ∂1 a, where α(gij, gjk, gkl) is the associator gij ◦ (gjk ◦ gkl) ⇐ (gij ◦ gjk) ◦ gkl, where ◦h is the horizontal composition of 2-morphisms.

• A 1-morphism from (θ,e g,e ea) to (θ, g, a) consists of

1M0 a 1-morphism A = (Ai): θ ← θe in F(tUi); ∗ ∗ 1M1 a 2-morphism f : g ◦ ∂0 A ⇐ ∂1 A ◦ ge in F(tUij); −1 ∗ ∗ ∗ −1 1M2 a higher coherence condition, (a ◦h id) ◦ (id ◦h ∂0 f) ◦ (∂2 f ◦h id) = ∂1 f ◦ (id ◦h ea) of 2-morphisms in F(tUijk).

• A 2-morphism from (A,e fe) to (A, f) consists of

5 2M0 a 2-morphism ω : A ⇐ Ae in F(tUi); ∗ ∗ 2M1 a higher coherence condition f ◦ (∂1 ω ◦h id) = (id ◦h ∂0 ω) ◦ fe. Then the (3, 1)-sheaf F + maps M ∈ M to the following 2-groupoid:

+ •F (M)0: an object is a pair ({Ui},P ), where {Ui} is a cover of M and P is an object in holim F(U(M)•);

+ •F (M)1: a 1-morphism from ({Uei}, Pe) to ({Ui},P ) is a common reﬁnement {Vi} of {Uei} and {Ui}, and a 1-morphism φ in holim F(V (M)•);

+ •F (M)2: a 2-morphism from ({Vei}, φe) to ({Vi}, φ) is a common refinement {Wi} of {Vei} and {Vi}, and a 2-morphism α in holim F(W (M)•). Moreover, ({Wfi}, αe) and ({Wi}, α) are identified if α and αe are identified on a further common refinement of {Wfi} and {Wi}. The plus construction for (2, 1)-presheaves is then a truncation of that of (3, 1)-sheaves viewing 1- groupoids as 2-groupoids with identity 2-morphisms. Let us explain it with a nice example.

p op Example 2.1. Given a Lie group G and its Lie algebra g, there is a (2, 1)-presheaf BGc : Mfd → Gpd sending U ∈ Mfd to the groupoid whose objects are trivial G-principal bundles U × G together with θ ∈ Ω1(U, g) and whose morphisms from (U × G, θe) to (U × G, θ) are gauge transformations g : U → G ∗ satisfying θ −Adgθe = −g θMC; and sending a morphism U → V to the functor between the corresponding groupoids induced by pullbacks of principal bundles and diﬀerential forms. Here θMC is the right invariant Maurer-Cartan form on G. It satisﬁes the following Maurer-Cartan equation 1 dθ − [θ , θ ] = 0. MC 2 MC MC g p Let us form the holim BGc (U(M)•) with respect to an open cover {Ui} of M ∈ Mfd. An object in p holim BGc (U(M)•) consists of 1 • Ui × G, θi ∈ Ω (Ui, g);

gij ∗ • gij : Uij → G, (Ui × G, θi) ←−− (Uj × G, θj), with θi − adgij θj = −gijθMC;

• compatibility condition gij ◦ gjk = gik on Uijk. Thus, we see that such an object gives us exactly the local data of a G-principal bundle with a connection p 1-form. A morphism in holim BGc (U(M)•) from (θei; geij) to (θi; gij) consists of ∗ • gi : Ui → G, θi − adgi θei = −gi θMC;

• compatibility condition gij · gj = gi · geij on Uij. This gives us exactly the local data of a gauge transformation preserving connections between the corresponding G-principal bundle glued by geij and gij.

3 (3, 1)-sheaf of string principal bundles

3.1 Finite dimensional model of Stringp(G) In this section, G is a finite dimensional compact Lie group. Let us first recall the finite dimensional 4 model of the Lie 2-group Stringp(G) built in [39] for a given class p ∈ H (BG•, Z). The idea is to realise Stringp(G) as a BU(1)-central extension of a Lie group G

BU(1) −→ Stringp(G) −→ G,

6 4 with the extension class p ∈ H (BG•, Z). Here BG• is the simplicial nerve of a Lie group G, and • H (BG•, Z) denotes the sheaf cohomology of the sheaf of Z-valued functions on BG•. Similarly, we use • • H (BG•,U(1)) and H (BG•, R) to denote the sheaf cohomology of U(1)-valued function and R-valued function respectively. Let us explain a bit the terminology here: a Lie 2-group is a diﬀerentiable stack equipped with a group structure (up to homotopy). For example, BU(1) is an abelian Lie 2-group. Here BU(1) denotes the stack presented by groupoid U(1) ⇒ pt. The multiplication m : BU(1) × BU(1) → BU(1) is induced by the multiplication of U(1). Notice that since U(1) is abelian, thus the U(1)-multiplication is a functor

(U(1) ⇒ pt) × (U(1) ⇒ pt) −→ (U(1) ⇒ pt).

For more details in the topic of (Lie) 2-groups and examples see e.g. [4][14, Sect 3.1]. In the case that 4 1 G = Spin(n), the generator of H (BG•, Z) is given by 2 p1, half of the Pontryagin class. The sheaf cohomology may be calculated by taking a hypercover of BG• and taking the Čech cohomology, as long as the hypercover is acyclic. This is a classical result. See [52, Proposition 2.4] for a concrete statement. See [23] and [52, Sect. 2] and references therein for deﬁnition and properties of the sheaf cohomology for simplicial objects. The cohomology on BG• is then equivalent to the group cohomology used in [39] originally coming from Segal and Brylinski [40, 41, 11]. In our case, as long as the cover of BG• on each layer G(•) is good, namely intersections are contractible, it is acyclic with respect to sheaves in our study. The short exact sequence of sheaves 0 → Z → R → U(1) → 0 gives us a long exact sequence of ≥1 n ∼ n+1 cohomology. Notice that H (BG•, R) = 0 for compact group G. Thus H (BG•,U(1)) = H (BG•, Z) for n ≥ 1. To build the ﬁnite dimensional model for the Lie 2-group Stringp(G), let us take a good simpli- (•) cial hypercover G for BG• and write the simplicial-Čech double complex whose total cohomology is • H (BG•,U(1)).

O O δ δ ˇ ˇ C(tG(3),U(1)) δ C(tG(3),U(1)) δ ... s / s,t / O O O δ δ δ

ˇ ˇ ˇ C(tG(2),U(1)) δ C(tG(2),U(1)) δ C(tG(2) ,U(1)) δ ... p / p,q / p,q,r / O O O O δ δ δ δ

(1) δˇ (1) δˇ (1) δˇ (1) δˇ C(tGα ,U(1)) / C(tGα,β,U(1)) / C(tGα,β,γ ,U(1)) / C(tGα,β,γ,δ,U(1)) / ... O O O O δ δ δ δ

0 id 0 id C(·,U(1)) / C(·,U(1)) / C(·,U(1)) / C(·,U(1)) / ... (3) 3 4 We take a representative (Θ, Φ, η, 0) of p ∈ H (BG•,U(1)) = H (BG•, Z), where

(3) (2) (1) Θ ∈ C(tGs ,U(1)), Φ ∈ C(tGp,q,U(1)), η ∈ C(tGα,β,γ ,U(1)). The last entry being 0 is implied by the closedness. To build up a Lie 2-group, we ﬁrst need to have an underlying Lie groupoid which presents the stack Stringp(G), and then establish a group structure “up to homotopy” on top of it. Here we follow the convention in [52, Section 2]. Our underlying Lie groupoid Γ[η] is a U(1)-extension of the Čech groupoid with respect to the cover (1) (1) (1) G , that is tGα,β × U(1) ⇒ tGα , together with source and target s(gα,β, a) = gα, t(gα,β, a) = gβ,

7 multiplication (gα,β, a)(gβ,γ , b) = (gα,γ , a + b − ηα,β,γ (g)), −1 ˇ identity e(gα) = (gα,α, 0), and inverse (gα,β, a) = (gβ,α, −a). Here δ(η) = 0 guarantees that the construction gives rise to a Lie groupoid structure. Now we build the multiplication for the 2-group structure on Γ[η], which should be a generalized morphism Γ[η]×2 −→ Γ(η). We realize the generalized morphism by a span of a Morita morphism and a usual morphism, Γ[η]×2 M.E.←− Γ2[η] −→m Γ(η),

2 (2) ×2 (2) where Γ [η] = G[1] × U(1) =⇒ G[0] , is similarly constructed as Γ[η], however, by the pullback Čech ∗ ∗ (2) ×2 (j) cocycle (d0(η), d2(η)) ∈ C(G[2] ,U(1) ). Here, and later, G[i] denotes the disjoint union of (i + 1)-fold intersections of the hypercover G(j). The natural projection Γ2[η] → Γ×2[η] is a Morita morphism, i.e., a morphism that gives arise to a Morita equivalence of Lie groupoids. The morphism Γ2(η) −→m Γ(η) is given by

m (v0, v1, a0, a1) −→ (d1(v0), d1(v1), a0 + a1 + Φ(v0, v1)).

It being a groupoid morphism is equivalent to the fact that δ(η) + δˇ(Φ) = 0. However the multiplication is not strictly associative, that is, the following diagram of diﬀerentiable stacks commutes up to a 2- morphism a, which is called an associator,

×3 m×id ×2 Stringp(G) / Stringp(G)

id×m m

×2 m Stringp(G) / Stringp(G).

×3 We now ﬁnd a suitable Lie groupoid presentation of Stringp(G) so that certain desired morphisms can be written as strict morphisms of Lie groupoids. Notice that there are three maps d0d0, d2d0, d2d2 : G×3 −→ G. Just like before, we take Γ3[η] to be the Lie groupoid constructed by the pullback cocycle ∗ ∗ ∗ (3) ×3 3 (3) ×3 (3) (d0d0) η, (d2d0) η, (d2d2) η ∈ C(G[3] ,U(1) ), that is, Γ [η] = G[1] ×U(1) =⇒ G[0] . Now the two ×3 id×m ×2 m ×3 m×id composed morphism m1 : Stringp(G) −−−→ Stringp(G) −→ Stringp(G) and m2 : Stringp(G) −−−→ ×2 m Stringp(G) −→ Stringp(G) are given by strict Lie groupoid morphisms:

∗ ∗ m1 : ((ws, wt), a0, a1, a2) 7−→ (d1d2(ws), d1d2(wt), a0 + a1 + a2 + d2Φ(ws, wt) + d0Φ(ws, wt)), ∗ ∗ m2 : ((ws, wt), a0, a1, a2) 7−→ (d1d1(ws), d1d1(wt), a0 + a1 + a2 + d1Φ(ws, wt) + d3Φ(ws, wt)).

3 In this model, the associator a : m2 =⇒ m1, is a map Γ [η]0 −→ Γ[η]1, given by

w0 7−→ (d1d2(w0), d1d2(w0), Θ(w0)). ˇ The naturality condition m1(r)a(s(r)) = a(t(r))m2(r) is equivalent to the equation δ(Φ) − δ(Θ) = 0. The pentagon condition for the associator is equivalent to the equation δ(Θ) = 0.

p+ 3.2 (3, 1)-sheaf BString(n)c of string principal bundles with connection data 1 4 Now we take G = Spin(n) and p to be 2 p1 ∈ H (BSpin(n)•, Z). We denote by String(n) the corresponding string group Stringp(G). p op The (3, 1)-presheaf of String(n)-principal bundles with connections BString(n)c : Mfd → 2Gpd is p constructed as following: ﬁrst of all, BString(n)c (U) is a 2-groupoid made up by the following data:

8 p • BString(n)c (U)0: an object is a triple U × String(n) → U, θ, B consisting of a locally trivial String(n)-principal bundle (U × String(n) → U) together with a 2-form B ∈ Ω2(U) and an so(n)- valued 1-form θ ∈ Ω1(U, so(n)).

p • BString(n)c (U)1: a 1-simplex

2 (g01,A01,ω01) U × String(n) → U, θ0,B0 ←−−−−−−−− U × String(n) → U, θ1,B1

2 consists of a generalized morphism g01 : U → String(n) given by a trivial bibundle Eg01 , a 1-form 1 2 2 A01 ∈ Ω (U) and a 2-form ω01 ∈ Ω (U) such that

2 2 ∗ B1 − B0 = ω01 + dA01, cs3(θ1) − cs3(θ0) = dω01, θ0 − adg¯01 θ1 = −g¯01θMC,

where cs3(θ) is the Chern-Simon 3-form associated to an so(n)-valued 1-form θ given by 1 cs (θ) = (θ, dθ) + (θ, [θ, θ]). (4) 3 3

g01 π Here (−, −) is a certain invariant symmetric bilinear form on so(n), and g¯01 : U −−→ String(n) −→ π Spin(n) is the composition of g01 with the natural projection String(n) −→ Spin(n). Composition is given by the multiplication in String(n):

2 2 2 2 (g01,A01, ω01) ◦ (g12,A12, ω12) = (g01 · g12,A01 + A12, ω01 + ω12).

p 2 • BString(n)c (U)2: a 2-simplex with edges (gij,Aij, ωij) for 0 ≤ i < j ≤ 2, or equivalently (in the 2 2 2 model of bicategory), a 2-morphism between (g01,A01, ω01)◦(g12,A12, ω12) and (g02,A02, ω02), is given by a pair (f, ω1) with f ∈ C∞(U, U(1)) and ω1 ∈ Ω1(U) such that

1 2 2 2 1 A12 − A02 + A01 = ω − d log f, ω12 − ω02 + ω01 = −dω .

3 Moreover, f gives rise to an isomorphism of bibundles g01 · g12 ⇐ g02.

(θ1,B1)

(g ,A , ω2 ) (g ,A , ω2 ) 01 01 01 (f, ω1) 12 12 12

(g ,A , ω2 ) (θ0,B0) 02 02 02 (θ2,B2).

φ p p Given a morphism U −→ V , the associated functor BString(n)c (V ) → BString(n)c (U) is given by pre-compositions and pullbacks of forms. p Now let us look at holim BString(n)c (U(M)•) for a cover {Ui} of M, here U(M)• given in (2) is the p nerve of the Čech groupoid associated to the cover {Ui}. An object in holim BString(n)c (U(M)•) consists of

2 1 • Ui × String(n) → Ui, Bi ∈ Ω (Ui), θi ∈ Ω (Ui, so(n));

1 2 2 • gij : Uij → String(n), Aij ∈ Ω (Uij), ωij ∈ Ω (Uij), such that

2 2 ∗ Bj − Bi = dAij + ωij, cs3(θj) − cs3(θi) = dωij, θi − adg¯ij θj = −g¯ijθMC; (5)

2This means that the bibundle as a right principle bundle is a trivial right principal bundle. See Lemma 3.4. 3Note that a U(1)-valued function on U provides an isomorphism of bibundles from U to String(n) via the map BU(1) → String(n). See also Lemma 3.4.

9 1 1 • fijk : Uijk → U(1), ωijk ∈ Ω (Uijk), such that fijk is an isomorphism gik ⇒ gij · gjk, and

ˇ 1 ˇ 2 1 (δA)ijk = ωijk − d log fijk, (δω )ijk = −dωijk. (6)

• a pentagon condition for 2-morphisms indicated by the following diagram,

(f ,ω1 ) 2 2 2 ijk ijk 2 2 (gij,Aij, ωij) ◦ (gjk,Ajk, ωjk) ◦ (gkl,Akl, ωkl) ks (gik,Aik, ωik) ◦ (gkl,Akl, ωkl) em 1 (fikl,ωikl)

(id◦ a,0,0) 2 h (gil,Ail, ωil) 1 (fijl,ωijl)

1 (fjkl,ω ) qy 2 2 2 jkl 2 2 (gij,Aij, ωij) ◦ (gjk,Ajk, ωjk) ◦ (gkl,Akl, ωkl) ks (gij,Aij, ωij) ◦ (gjl,Ajl, ωjl), (7) where a is the associator of the string group String(n), and ◦h is the horizontal composition of 2-morphisms, noticing that (gij ◦ gjk) ◦ gkl and gij ◦ (gjk ◦ gkl) are composed 1-morphisms (gij ,gjk,gkl) ×3 Uijk −−−−−−−−→ String(n) → String(n). According to Lemma 3.5, the 2-morphism id ◦h a is given by U(1)-valued functions Fijkl : Uijkl → U(1), which converges to a class in the Čech co- ˇ 3 1 homology group H (M, Z) determined by the extension class 2 p1. Thus the above diagram says exactly δfˇ = F, δωˇ 1 − δdˇ log f = 0, δdωˇ 1 = 0. (8) The latter two equations are implied by (6).

p 2 1 A 1-morphism in holim BString(n)c (U(M)•) from (Ui × String(n) → Ui, θei, Bei; geij, Aeij, ωeij; feijk, ωeijk) 2 1 to (Ui × String(n) → Ui, θi,Bi; gij,Aij, ωij; fijk, ωijk) consists of

1 2 2 2 p • gi : Ui → String(n), Ai ∈ Ω (Ui), ωi ∈ Ω (Ui), such that (gi,Ai, ωi ) ∈ BString(n)c (tiUi)1 is a 1-morphism between

2 (gi,Ai,ωi ) (Ui × String(n) → Ui, θi,Bi) ←−−−−−− (Ui × String(n) → Ui, θei, Bei),

that is, 2 2 ∗ Bei − Bi = ωi + dAi, dωi = cs3(θei) − cs3(θi), θi − adg¯i θei = −g¯i θMC; (9)

1 p • an element (fij, ωij) ∈ BString(n)c (tijUij)2 which provides a 2-morphism making the following diagram 2-commutative 2 (gj ,Aj ,ωj ) (θj,Bj) o (θej, Bej)

2 2 (gij ,Aij ,ω ) (g ,A ,ω ) ij eij eij eij (θ ,B )(θ , B ), i i o 2 ei ei (gi,Ai,ωi ) that is

f ij 1 2 2 2 2 1 gij · gj ks gi · geij,Aij + Aj − (Ai + Aeij) = ωij − d log fij, ωij + ωj − (ωeij + ωi ) = −dωij. (10)

10 • a higher coherence condition between 2-morphisms

(θk,Bk) o (θek, Bek)

z z (θj,Bj) o (θej, Bej)

$ $ (θi,Bi)(o θei, Bei)

which gives us the following equations4

−1 ˇ 1 1 ˇ 1 ˇ fijkfeijk = (δf..)ijk, ωeijk − d log feijk − ωijk + d log fijk = −(δω..)ijk + (δd log f..)ijk,

p 2 1 2 1 A 2-morphism in holim BString(n)c (U(M)•) from (gei, Aei, ωei ; feij, ωeij) to (gi,Ai, ωi ; fij, ωij) consists of

1 p 2 2 fi • an element (fi, ωi ) ∈ BString(n)c (tUi)2 from (gei, Aei, ωei ) to (gi,Ai, ωi ), that is, gi ks gei , and

1 2 2 1 Ai − Aei = −d log fi + ωi , ωi − ωei = −dωi ;

• a coherence condition held on Uij,

2 1 1 2 (fij, ωij) ◦ (fi, ωi ) = (fj, ωj ) ◦ (feij, ωeij), that is5, −1 −1 1 1 ˇ 1 fijfeij = fjfi , ωij − ωeij = (δω. )ij.

p + op Then (BString(n)c ) : Mfd → 2Gpd is a (3, 1)-sheaf consists of p+ • BString(n)c (M)0: an object is a pair ({Ui},Pc), where {Ui} is an open cover of M and Pc is an p element in holim BString(n)c (U(M)•)0; p+ • BString(n)c (M)1: a 1-morphism between ({Ui},Pc) and ({Uei}, Pec) is a pair consisting of a com- p mon refinement {Vi} of {Ui} and {Uei} and an element φc ∈ holim BString(n)c (V (M)•)1; p+ • BString(n)c (M)2: a 2-morphism between ({Vi}, φc) and ({Vei, φec}) consists of a common refine- p ment {Wi} of {Vi} and {Vei} and an element αc ∈ holim BString(n)c (W (M)•)2. Moreover, ({Wi}, αc) and ({Wfi}, αec) are identified if αc and αec agree on a further common refinement of {Wi} and {Wfi}. p Remark 3.1. For simplicity, we call an element Pc ∈ holim BString(n)c (U(M)•)0 a string data. The construction of our string (pre)sheaf works for a general compact Lie group G with the extension class p a multiple of p1, as long as we adjust the coefficient in the front of the Chern-Simon 3-form by this multiple also.

4These two equations are deduced from the coherence condition between g’s and A’s, the one for ω2’s can be implied by the second equation in (10). 5The same as before, the two equations are deduced from the coherence condition between g’s and A’s, and the one coming from ω2’s can be implied by the second equation.

11 3.3 Lifting Theorem and Comparison As we state in the introduction, there are already several ways to grasp the concept of string structure. Redden’s string class is probably the most accessible and concise, while Waldorf’s method includes connection data and makes it easy and natural to locate the integrity of the string class. The reason to develop yet another way here, is to connect with concepts involving differential forms, such as Courant algebroids, descent equations, and Deligne cohomology. We found it also much easier in this language to relate to physics literatures, such as [22]. Then, we own readers a justification. The direct comparison to previous methods might be a wrong approach to see the nature of the problem since both sides (especially our side) involves heavy machinery. We remark (Remark 3.7, 3.9, 3.10, 3.11) carefully on links of these concepts, and focus ourselves on the proof of lifting theorem from the viewpoint of Stolz-Teichner for justification. p+ Let BGc be the (2, 1)-sheaf of G-principal bundles with connections. We take the model BGc for p it. Thus comparing the construction of BString(n)c with Example 2.1, we see that there is a natural p+ π projection BString(n)c −→ BSpin(n)c, by forgetting higher data. More precisely, given an object

2 1 p+ Pc = (tiUi; Ui × String(n) → Ui, θi,Bi; gij,Aij, ωij; fijk, ωijk) ∈ BString(n)c ,

2 1 where (θi,Bi; gij,Aij, ωij; fijk, ωijk) satisﬁes equations (5), (6) and (8), since the isomorphism of bibundles gij between gij ·gjk ⇐ gik is given by a U(1)-function fijk, the projected Spin(n)-valued function, g¯ij : Uij −−→ String(n) → Spin(n), satisﬁes strict cocycle condition g¯ij · g¯jk =g ¯ik. This gives us a Spin(n)-principal bundle P¯. Furthermore, equation (5) implies that θi provides a connection on P¯.

P¯c Theorem 3.2. An object M −→ BSpin(n)c in BSpin(n)c over a ﬁne enough good cover {Ui} of M lifts Pc p+ p+ to an object in M −→ BString(n)c in BString(n)c

p+ BString(n)c 9

P¯c M / BSpin(n)c,

1 ¯ ¯ ¯ if and only if 2 p1(P ) = 0, where P is the Spin(n)-principal bundle that Pc glues to. Remark 3.3. This theorem says that as long as a good cover is fine enough, the obstruction for a Spin(n) ¯ 1 ¯ data Pc to lift to a String(n) data is 2 p1(Pc). Since we may always find fine enough good covers, this p theorem justifies that our construction BString(n)c is indeed reasonable to be called the (3, 1)-presheaf of String(n)-principal bundles. We first prove some technical lemmas: Lemma 3.4. Given a function g¯ : U → G, one may always lift it to a morphism g : U → String(G).

(1) (1) Proof. A morphism g : U → String(G) is given by a bibundle E which is a tGαβ × U(1) ⇒ tGα principal bundle over U. We know that the underlying morphism g¯ : U → G of g is given by the bibundle (1) (1) U ×g,G,¯ pr tGα , and E is a trivial U(1)-bundle over it, that is E = U ×g¯ij ,G,pr tGα × U(1). Suppose that the action is given by

0 0 0 (1) (x, gα, a) · (gαβ, a ) := (x, gβ, a + a + λαβ(x)), for (x, gα, a) ∈ E, and (gαβ, a ) ∈ Gαβ × U(1),

for a certain function λαβ : U → U(1). The associativity of the action is equivalent to the fact that ˇ 2 (1) ∗ (δλ)αβγ = ηαβγ (¯g(x)). But H (U ×g,G,¯ pr tGα ,U(1)) = 0, thus 2-cocycle g¯ η is always exact. Therefore we may always ﬁnd such λ.

12 (•) We endow a simplicial hypercover G of BG•–the nerve of G. Suppose that the extension class 3 ∼ 4 p ∈ H (BG•,U(1)) = H (BG•, Z) is represented by the U(1)-valued 3-cocycle (Θ, Φ, η, 0) supported on this cover. The last entry being 0 is implied by the closedness. Notice that g¯ij : Uij → G extends to a simplicial morphism g¯(•) from the simplicial nerve U(M)• of the covering groupoid tUij ⇒ tUi to BG•.

On each simplicial level U(M)k = tUi0i1...ik , we endow it with the pullback cover of the one on BGk pulled back by g¯(k). We may always start with a ﬁne enough cover {Ui} so that g¯ij(Uij) is either entirely in Gα or does not intersect Gα. Thus we may assume that both {Ui} and the pullback covers are good. Then • ∼ • the simplicial-Čech double complex (11) calculates the cohomology H (U(M)•,U(1)) = Hˇ (M,U(1)). ∗ ¯ ¯ We denote the pullback cocycle g¯(•)(Θ, Φ, η, 0) by (Θ, Φ, η,¯ 0) and it is a cocycle in double complex (11) 3 representing a class p(P¯c) ∈ Hˇ (M,U(1)).

O O δ δ ˇ ˇ C(tU ,U(1)) δ C(tU ,U(1)) δ ... ijkl;s / ijkl;s,t / O O O δ δ δ ˇ ˇ ˇ C(tU ,U(1)) δ C(tU ,U(1)) δ C(tU ,U(1)) δ ... ijk;p / ijk;p,q / ijk;p,q,r / O O O O δ δ δ δ

δˇ δˇ δˇ δˇ C(tUij;α,U(1)) / C(tUij;α,β,U(1)) / C(tUij;α,β,γ ,U(1)) / C(tUij;α,β,γ,δ,U(1)) / ... O O O O δ δ δ δ

0 id 0 id C(tUi,U(1)) / C(tUi,U(1)) / C(tUi,U(1)) / C(tUi,U(1)) / ... (11)

Lemma 3.5. The 2-morphism (id ◦h a, 0, 0) in diagram (7) is given by U(1)-valued functions Fijkl : 1 ¯ Uijkl → U(1). Moreover, if the cover is ﬁne enough, Fijkl’s give rise to a cocycle representing 2 p1(Pc) ∈ Hˇ 3(M,U(1)).

Proof. We continue to use the notation and a ﬁne enough cover {Ui} given just before this lemma. For 2 ˇ ˇ us now G = Spin(n). Since Hˇ (tUij,U(1)) = 0, (δη¯) = 0 implies that η¯ = −δλ. We continue this ≥0 ≥0 tic-tac-toe procedure, since Hˇ (tUijk,U(1)) = Hˇ (tUijkl,U(1)) = 0, we have

η¯ = −δλ,ˇ Φ¯ = δλ + δϕ,ˇ Θ¯ = δϕ + F, (12) and F is a function Uijkl → U(1). A calculation shows that the bibundle of (gij ◦ gjk) ◦ gkl is

(3) ×3 (3) ×3 ×3 Uijkl ×G tGs × (U(1)) ×tGα tGαβ × U(1)/ t Gs,t × (U(1)) , where the action is given by

0 0 0 (xijkl, ws, a1, a2, a3, gα,β, a) · (ws,t, a1, a2, a3) 0 0 0 =(xijkl, wt, a1 + a1 + λα1,γ1 (xij), a2 + a2 + λα2,γ2 (xjk), a3 + a3 + λα3,γ3 (xkl), gγ,β, 0 0 0 ∗ ∗ a − a1 − a2 − a3 − d2Φ(ws,t) − d0Φ(ws,t) − η¯α,β,γ (xil)).

This bibundle is isomorphic to the bibundle Uijkl ×G tGα × U(1) through

ψ [(xijkl, ws, a1, a2, a3, gα,β, a)] −→ [(xijkl, gβ, a + a1 + a2 + a3 + ϕp0 (v0) + ϕp2 (v2) + λα,β(xijkl))],

13 where the right action of tGα,β × U(1) ⇒ tGα is given by

0 0 (xijkl, gβ, a) · (gβ,β0 , a ) = (xijkl, gβ0 , a + a + λβ,β0 (x)).

The morphism ψ is well-defined thanks to the second equation in (12), and it is a bibundle isomorphism thanks to the first equation in (12). The bibundle of gij ◦ (gjk ◦ gkl) is given by exactly the same form but the quotient is given by a different action,

0 0 0 (xijkl, ws, a1, a2, a3, gα,β, a) · (ws,t, a1, a2, a3) 0 0 0 =(xijkl, wt, a1 + a1 + λα1,γ1 (xij), a2 + a2 + λα2,γ2 (xjk), a3 + a3 + λα3,γ3 (xkl), gγ,β, 0 0 0 ∗ ∗ a − a1 − a2 − a3 − d1Φ(ws,t) − d3Φ(ws,t) − η¯α,β,γ (xil)).

Similarly, this bibundle is also isomorphic to the same bibundle Uijkl ×G tGα × U(1) through

ψ0 [(xijkl, ws, a1, a2, a3, gα,β, a)] −→ [(xijkl, gβ, a + a1 + a2 + a3 + ϕp1 (v1) + ϕp3 (v3) + λα,β(xil))].

The 2-morphism id ◦h a is then to add Θ on the last U(1) component, and is explicitly given by

(xijkl, ws, a1, a2, a3, gα,β, a) 7→ (xijkl, ws, a1, a2, a3, gα,β, a − Θ(ws)),

before quotient. This map is equivariant with respect to the above two actions because δΦ = δˇΘ, thus it descends to the quotients. Under the isomorphism ψ and ψ0, this 2-morphism is then given by

0 −1 ψ (id ◦h a(ψ (xijkl, gβ, a))) = (xijkl, gβ, a + Fijkl(x)),

guaranteed by the last equation of (12). Moreover, [(F, 0, 0, 0)] = [(Θ¯ , Φ¯, η,¯ 0)] = 1 p (P¯ ) ∈ Hˇ 3 (M,U(1)). 2 1 c U• 1 ¯ Thus F is a representative of 2 p1(Pc). Now we are ready to prove Theorem 3.2. ¯ p Pc Proof. If there is a lifting object Pc ∈ BString(n)c over an object M −→ BSpin(n)c over {Ui} in BSpinc, we write 2 1 Pc = (tiUi; Ui × Stringp(n) → Ui, θi,Bi; gij,Aij, ωij; fijk, ωijk), 2 1 where (θi,Bi; gij,Aij, ωij; fijk, ωijk) satisﬁes equations (5), (6) and (8). Then (8) and Lemma 3.5 implies 1 ¯ that 2 p1(Pc) = 0. P¯c For the other direction, we ﬁrst do some preparation: given an object M −→ BSpin(n)c over a good cover {U } of M in BSpin(n) , we take a U(1)-cocycle F representing 1 p (P¯ ) ∈ Hˇ 3 (M,U(1)) = i c 2 1 c U• H3(M,U(1)). Let d log 1 d 2 d d m Dm = U(1) −−−→ Ω −→ Ω −→ ... −→ Ω (13) • be the Deligne sheaf of depth m. Recall that the Deligne cohomology H (M,D3) is then the limit of the total cohomology of the following double complex taking over all covers {Ui} of M,

d log 1 d 2 d 3 ˇ C(U(M)•,U(1)) −−−→ Ω (U(M)•) −→ Ω (U(M)•) −→ Ω (U(M)•), δ .

• π Then the general theory of Deligne cohomology tells us that there is a surjective morphism H (M,D3) −→ H•(M,U(1)) given by forgetting the part of diﬀerential forms of a Deligne cocycle. Moreover, there is a ¯ 3 d 4 1 2 3 3 3 3 morphism H (M,D3) −→ Ωcl(M) to closed 4-forms by [(F, ω , ω , ω )] 7→ dω . Here notice that ω = (ωi ) 3 is made up by local 3-forms on each Ui. However, dωi ’s glue together to a closed global 4-form, which

14 we denote by dω3, and it is independent of the choice of the Deligne cocycle. The above two morphisms ﬁt into the following commutative diagram:

¯ 3 d 4 4 H (M,D3) / Ωcl(M) / HdR(M, R) (14) 8 ⊗ π R H3(M,U(1)) / H4(M, Z)

3 3 Since the natural morphism H (M,D3) → H (M,U(1)) is surjective, we lift F to a Deligne cocycle (F, ω1, ω2, ω3), that is

δωˇ 1 − d log F = 0, δωˇ 2 + dω1 = 0, δωˇ 3 − dω2 = 0. (15)

3 ¯ Now we adjust ωi to be cs3(θi), where θi is the connection data with respect to the cover {Ui} in Pc. 3 1 Both dωi and dcs3(θi) give to closed global 4-forms, and both represent the deRham classes 2 p1 ⊗ R. 3 3 2 Thus ωi −cs3(θi) = γ +dβi, where γ ∈ Ω (M) and βi ∈ Ω (Ui). Then it is easy to verify that the Deligne ˇ 1 2 3 class (F, ω1, ω2 + δβ, cs3(θi)) is a lift of F . Thus we can begin with a Deligne cocycle (F, ω , ω , ω ) with 3 ωi = cs3(θi). ¯ 1 ¯ Now we construct a lift of Pc with respect to the cover {Ui} under the condition 2 p1(Pc) = 0. Fix (•) a good cover G of BSpin(n)• as in the construction of String(n) in Subsection 3.1. Refining {Ui} if (1) (1) necessary, we may assume that g¯ij(Uij) either falls entirely into Gα or does not intersect Gα . Then the condition in Lemma 3.4 is naturally fulfilled. Thus, there is no obstruction to lift the transition functions ¯ 1 ¯ ˇ g¯ij for Pc to gij : Uij → String(n). Since 2 p1(Pc) = 0, we may take a primitive f of F , that is F = δf. ˇ 1 1 ˇ 1 Since d log F = δω , ω = δA + d log f for some A = (Aij) ∈ Ω (tUij). We continue such a tic-tac-toe process, and find F = δf,ˇ ω1 = δAˇ + d log f, ω2 = δBˇ − dA, (16) 1 2 for f = (fijk) ∈ U(1)(tUijk), A = (Aij) ∈ Ω (tUij), and B = (Bi) ∈ Ω (tUi). Both gij ◦ gjk and gik are morphisms from Uijk to String(n). They are presented by isomorphic bibundles from the discrete groupoid Uijk ⇒ Uijk to Γ[η] with a similar construction to that of ψ in Lemma 3.5. Then as in the proof of Lemma 3.4, we see that a U(1)-valued function fijk on Uijk serves as a 2-morphism gik ⇒ gij ◦ gjk because U(1) ⇒ pt is a subgroupoid of Γ[η] and sits in the center of it. Thus, (tiUi; Ui × String(n) → 2 1 ¯ Ui, θi,Bi; gij,Aij, ωij; fijk, ωijk) is a lift of Pc.

As shown in [48], if a Spin(n)-principal bundle P¯ admits string classes then the possible choices of the string classes form a torsor of H3(M, Z). Then later in [50], the author further showed that for a fixed Chern-Simon 2-gerbe over P¯, the choices of trivialisations modding out isomorphisms correspond exactly to string classes on P¯. p We see that inside an object Pc ∈ BString(n)c , the determining information, is a covering {Ui} 1 2 3 together with (θi,Bi; gij,Aij, fijk), other terms (Fijkl, ωijk, ωij, ωi = cs3(θi)) are determined by these 1 2 3 1 ¯ terms through (16). These terms (Fijkl, ωijk, ωij, ωi = cs3(θi)) representing a refinement of 2 p1(Pc) ∈ 3 H (M,U(1)), may be viewed as the information for a given Chern-Simon 2-gerbe over P¯c and its connections. Thus, after adding connection data inside, we may ask ourselves again how many string data ¯ 1 2 3 lift Pc ∈ BSpin(n)c if we fix a choice of the cocycle (Fijkl, ωijk, ωij, ωi = cs3(θi)). 2 2 ∼ 3 The Deligne cohomology group H (M,D2) may be viewed as a refinement of H (M,U(1)) = H (M, Z),

15 where D2 is the Deligne sheaf deﬁned in (13). We have the following diagram (see also (14))

¯ 2 ¯ π3 2 ∼ 3 ker π2 ∩ ker d = ker π3 / H (M,D3) = ker d / H (M,U(1)) = H (M, Z) 4 π2

2 ker π2 / H (M,D2)

d¯

3 Ωcl(M)

Since we have

3 2 2 3 2 H (M, Z) = H (M,D3)/ ker π3 ,→ H (M,D2)/ ker π3 → H (M, Z) = H (M,D2)/ ker π2, and H3(M, Z) may be viewed naturally both as a subgroup and a quotient of our group, we show that 2 diﬀerent lifts of P¯c modding out isomorphisms is a torsor of H (M,D2)/ ker π3 in the next theorem.

P¯c Theorem 3.6. Given an object M −→ BSpin(n)c over a good cover {Ui} of M in BSpin(n)c, and a 1 2 1 ¯ 3 fixed Deligne cocycle (Fijkl, ωijk, ωij, cs3(θi)) representing a refinement of 2 p1(Pc) ∈ H (M,U(1)), let us 1 2 denote the set of all possible Pc’s lifting P¯c with fixed (Fijkl, ω , ω , cs3(θi)) by S 1 2 . ijk ij (Fijkl,ωijk,ωij ,cs3(θi)) 2 Then the Deligne cohomology group H (M,D2) acts on

S 1 2 /1-morphisms. (Fijkl,ωijk,ωij ,cs3(θi))

2 This action descends to the quotient H (M,D2)/ ker π3 and makes S 1 2 /1-morphisms a (Fijkl,ωijk,ωij ,cs3(θi)) 2 (H (M,D2)/ ker π3)-torsor.

h h h 2 Proof. Given a cocycle (f ,A ,B ) representing an element in H (M,D2), it acts on S 1 2 (Fijkl,ωijk,ωij ,cs3(θi)) by h h h ·(f ,A ,B ) h h h (Ui, θi,Bi; gij,Aij, fijk) −−−−−−−−→ (Ui, θi,Bi + Bi ; gij,Aij + Aij, fijk + fijk). (17) h h h If (f ,A ,B ) = D(ϕ, α), then the above two elements in S 1 2 are connected by an iso- (Fijkl,ωijk,ωij ,cs3(θi)) 2 morphism (1, −α, 0; ϕ, 0). Thus (17) gives rise to an action of H (M,D2) on S 1 2 /1-morphisms. (Fijkl,ωijk,ωij ,cs3(θi)) Now we prove that the action of a cocycle (f h,Ah,Bh) induces an isomorphism if and only if h h h ¯ [(f ,A ,B )] ∈ ker π2 ∩ ker d. This then will complete the proof of our statement. First of all, the isomorphism induced by (f h,Ah,Bh) is possibly given through another ﬁner cover {Vi}. However as we may always pull back our cocycle to Vi, we might as well assume that Vi = Ui. By h h h 2 1 a direct calculation, (f ,A ,B ) induces an isomorphism (1,Ai, ωi ; fij, ωij), if and only if

h 2 2 h ˇ 1 ˇ 2 1 ˇ h −1 Bi = ωi + dAi, dωi = 0,Aij = δ(A.)ij − ωij + d log fij, δω . = −dωij, (δf..)ijk = (fijk) . (18)

h h h h h Thus one direction is clear. If (f ,A ,B ) induces an isomorphism, then dBi = 0 and f is a coboundary, h h h h h h which exactly shows that d[(f ,A ,B )] = 0 and π2([(f ,A ,B )]) = 0 respectively. h h ˇ h h For the other direction, if dBi = 0 and fijk = (δf..)ijk for some f.., then (1, 0,Bi ; fij, −Aij +d log fij) gives us a desired isomorphism.

Remark 3.7. If we forget the connection data inside a string data Pc and only remember (Ui; gij; fijk), 2 3 then the action of H (M,D2) simpliﬁes to that of H (M, Z) through π2. Thus we recover the structure of the torsor in [48] for the string structures over a Spin(n)-principal bundle via the projection 2 3 H (M,D2)/ ker π3 → H (M, Z).

16 ˇ For a string data Pc = (Ui × String(n), θi,Bi; gij,Aij; fijk), we see that δ(cs3(θi) − dBi) = 0 by (5), thus {cs3(θi) − dBi} give rise to a global 3-form H on M. We deﬁne H to be the curvature of Pc. By (9), we see that isomorphic string data over the same P¯c has the same curvature. Notice that h h h h when (f ,A ,B ) acts on Pc, the curvature is changed by dBi which glues to a global closed 3-form ¯ h h h 3 d(f ,A ,B ) ∈ Ωcl(M). Thus we have the following corollary,

¯ Pc 1 2 Corollary 3.8. Given an object M −→ BSpin(n)c over M, and a Deligne cocycle (Fijkl, ωijk, ωij, cs3(θi)), ¯ 1 2 ¯ the curvatures of all possible Pc’s lifting Pc with ﬁxed (Fijkl, ωijk, ωij, cs3(θi)) form a torsor of im d. Remark 3.9. We notice the following commutative diagram

2 3 H (M,D2) / H (M, ) (19) π2 Z

d¯ ⊗R

3 3 Ωcl(M) / / H (M, R).

One may interpret im d¯ as “integral” forms. Certainly, d¯ is not always surjective. Remark 3.10. We conjecture that when we glue the local data of a string data, we will obtain an S1-gerbe over the underlying Spin(n)-principal bundle P¯. This gives us the access to Redden’s string class. For this problem, a possible way is to apply the descent for n-bundles of Wolfson [53, Theorem 5.7] to realize the gluing process. The difficulty would lie further on gluing differential forms to obtain an integral form which presents the string class. Notice that already in Redden’s thesis, there was no explicit formula to adjust to an integral form for the string class. We thus leave it for future investigation. Remark 3.11. To compare thoroughly with Waldorf’s method using trivialization of Chern-Simons 2- gerbe is neither a simple task. Both definitions use heavy machineries, one with bundle gerbe theory, and the other with stack theory. Nevertheless, some traces of equivalence are rather visible6. In [50, Definition 2.2.1], if we take Y to be tiUi, then bundle gerbe S characterised by a closed 3-form, over each Ui, corresponds to 2-form Bi in a string data, since a closed form is locally exact. Similarly the 2 1 bundle gerbe P corresponds to ωij, isomorphism A corresponds to Aij, M corresponds to ωijk, and σ corresponds to fijk. Then various coherence conditions correspond to our descent equations for differential forms.

p+ 4 (2, 1)-sheaf TCc of transitive Courant algebroids with connections

The notion of a Courant algebroid was introduced in [30]. See also [36, 37, 38, 45] for various other aspects of Courant algebroids. Deﬁnition 4.1. A Courant algebroid is a vector bundle C together with a bundle map ρ : C −→ TM, a nondegenerate symmetric bilinear form h−, −i, and an operation −, − : Γ(C) × Γ(C) −→ Γ(C) such that for all e1, e2, e3 ∈ Γ(C), the following axioms hold: J K (i) (Γ(C), −, − ) is a Leibniz algebra; J K 1 (ii) h e1, e1 , e2i = 2 ρ(e2) he1, e1i; J K (iii) ρ(e1) he2, e3i = h e1, e2 , e3i + he2, e1, e3 i. J K J K 6Here we thank Konrad Waldorf for very helpful conversations.

17 A Courant algebroid (C, −, − , h−, −i , ρ) is called transitive if ρ is surjective, that is, im ρ = TM. A transitive Courant algebroidJ is anK extension of a transitive Lie algebroid. However, not every transitive Lie algebroid A admits such a Courant extension. The obstruction is given by the ﬁrst Pontryagin class [9, 17, 43]. See also [31, 49] for more details about transitive Courant algebroids. In this section, we p introduce the (2, 1)-presheaf TCc of transitive Courant algebroids with connections, and we use the plus p+ construction to sheaﬁfy it to a (2, 1)-sheaf TCc . We then reinterpret the extension obstruction as the lifting obstruction, p+ TCc (20) =

p+ M / TLc . p p+ Here TLc and TLc are the (2, 1)-presheaf and (2, 1)-sheaf of transitive Lie algebroids with connections respectively (see appendix A.1). Notice how similar it is to the lifting story on the string side. Given a transitive Courant algebroid, we have the following two short exact sequences:

ρ 0 −→ ker ρ −→ C −→ TM −→ 0, (21) ρ0 0 −→ (ker ρ)⊥ −→ ker ρ −→ G −→ 0, (22)

where G = ker ρ/(ker ρ)⊥ is a Lie algebra bundle, whose ﬁber is isomorphic to a quadratic Lie algebra (g, (−, −)g). We will also use (−, −)g to denote the ﬁberwise metric on G.A connection of a transitive Courant algebroid C consists of the following data: • an isotropic splitting s : TM −→ C of the short exact sequence (21);

• a splitting σs : G −→ ker ρ of the short exact sequence (22) that is orthogonal to s(TM) in C, i.e. hs(X), σs(a)i = 0 for all X ∈ Γ(TM) and a ∈ Γ(G).

In [17], the authors show that splittings s and σs always exist. A connection gives rise to an isomorphism C =∼ TM ⊕ G ⊕ T ∗M between vector bundles. Transferring the Courant algebroid structure on C to ∗ ∗ T T TM ⊕G⊕T M, we obtain the transitive Courant algebroid (TM ⊕G⊕T M, −, − ∇,R,H , h−, −i , prTM ), which is determined by a connection ∇ on G, a 2-form R ∈ Ω2(M, G) andJ a 3-formK H ∈ Ω3(M), which T obey a set of conditions given in [17, Propositoin 2.2]. Here the bracket −, − ∇,R,H and the pairing T J K h−, −i are deﬁned by

T X + a + ξ, Y + b + η ∇,R,H = [X,Y ] + ∇X b − ∇Y a + [a, b]G + R(X,Y ) J K +LX η − iY dξ + P (a, b) − 2Q(X, b) + 2Q(Y, a) + H(X,Y ), (23) 1 hX + a + ξ, Y + b + ηiT = ξ(Y ) + η(X) + (a, b)g, (24) 2 where P : Γ(G) ⊗ Γ(G) −→ Ω1(M) and Q : X(M) ⊗ Γ(G) −→ Ω1(M) are given by

g P (a, b)(Y ) = 2(b, ∇Y a) , (25) Q(X, a)(Y ) = (a, R(X,Y ))g. (26)

In particular, if G is the trivial bundle M × g and the connection is given by ∇X a = X(a), we obtain the standard transitive Courant algebroid structure on TM ⊕ (M × g) ⊕ T ∗M with the Courant bracket given by T i X + a + ξ, Y + b + η S = [X,Y ] + X(b) − Y (a) + [a, b]g + LX η − Y dξ + P(a, b), (27) J K where P : Γ(M × g) ⊗ Γ(M × g) −→ Ω1(M) is given by

P(a, b)(Y ) = 2(b, Y (a))g.

18 For simplicity, for an object U ∈ Mfd, we write

∗ TgU := TU ⊕ (U × g) ⊕ T U. According to [17, Proposition 2.7], automorphisms of the standard transitive Courant algebroid are given as follows.

T T Corollary 4.2. An automorphism of the standard transitive Courant algebroid (TgM, −, − S , h−, −i , prTM ), J 1K 0 0  T T where −, − S and h−, −i are given by (27) and (24) respectively, is of the form  φ τ 0 , J K β −2φ?τ 1 where τ is an orthogonal automorphism of the bundle of quadratic Lie algebras M × g and φ : TM −→ M × g and β : TM −→ T ∗M are bundle maps satisfying the following compatibility conditions: 1 β(X)(Y ) + β(Y )(X) + (φ(X), φ(Y ))g = 0, (28) 2 τ(X(b)) − X(τ(b)) − [φ(X), τ(b)]g = 0, (29) 1 dφ + [φ, φ] = 0, (30) 2 g LX (β(Y )) − iY d(β(X)) − β([X,Y ]) + P(φ(X), φ(Y )) = 0. (31)

Here φ? : M × g −→ T ∗M is deﬁned by

φ?(a)(X) = (a, φ(X))g, ∀a ∈ Γ(M × g),X ∈ Γ(TM). (32)

p op There is a (2, 1)-presheaf of transitive Courant algebroids with connections TCc : Mfd → Gpd, where Mfdop is the opposite category of Mfd, and Gpd is the 2-category of (discrete) groupoids and groupoid morphisms. p For an object U ∈ Mfd, the groupoid TCc (U) is made up by the following data:

p T T 1 • TCc (U)0: an object is a 6-tuple (TgU, −, − S , h−, −i , prTU , θ, B), where θ ∈ Ω (U, g) is a g- 2 J K T T valued 1-form, B ∈ Ω (U) is a 2-form, and −, − S and h−, −i are given by (27) and (24) J K respectively. We will simply denote an object by (TgU, θ, B) in the sequel.

p • TCc (U)1: a 1-morphism from (TgU, θ,e Be) to (TgU, θ, B) is an automorphism of the standard tran-  1 0 0  T T sitive Courant algebroid (TgU, −, − S , h−, −i , prTU ) given by the matrix  φ τ 0  J K β −2φ?τ 1 such that

θ(X) = τθe(X) + φ(X), (33) g g ? iX (Be − B) = β(X) + (θ, θ(X)) − (θ,e θe(X)) − 2φ τ(θe(X)), (34)

cs3(θe) − cs3(θ) = d(Be − B). (35)

g Here the Chern-Simon 3-form cs3(θ) of θ is a 3-form on U deﬁned by (4) using bilinear form (−, −) . The composition of 1-morphisms is simply the matrix multiplication.

p p p Then for a morphism ϕ : U → V in Mfd, the associated functor TCc (ϕ): TCc (V ) → TCc (U) is induced by pulling back forms.

p Take an open cover {Ui} of M ∈ Mfd. An object in holim TCc (U(M)•) consists of p • an object t(TgUi, θi,Bi) in TCc (tUi)0,

19  1 0 0  p • a 1-morphism Λij =  φij τij 0  in TCc (tUij)1 from (TgUij, θj|Uij ,Bj|Uij ) to (TgUij, θi|Uij ,Bi|Uij ). ? βij −2φijτij 1 This implies that

θi(X) = τijθj(X) + φij(X), (36) i g g ? X (Bj − Bi) = βij(X) + (θi, θi(X)) − (θj, θj(X)) − 2φijτij(θj(X)), (37)

cs3(θj) − cs3(θi) = d(Bj − Bi). (38)

• compatibility conditions ΛijΛjk = Λik on Uijk, which are equivalent to the following equations

φij + τijφjk = φik,

τijτjk = τik, ? βij − 2φ τijφjk + βjk = βik, ? ? ? −2φijτijτjk − 2φjkτjk = −2φikτik.

T T p Deﬁnition 4.3. An object Cc = t(TgUi, −, − S , h−, −i , prTU θi,Bi); φij, τij, βij in holim TCc (U(M)•) J K i is called a transitive Courant data. For simplicity, we also denote Cc by (θi,Bi; φij, τij, βij) when there is no confusion. p p We describe the objects and morphisms of holim TCc (U(M)•) of the (2, 1)-presheaf TCc in the following two propositions.

Proposition 4.4. A transitive Courant data Cc = (θi,Bi; φij, τij, βij) gives rise to a transitive Courant algebroid (C, −, − , h−, −i , ρ) with a connection. J K Proof. Since ΛijΛjk = Λik, TgUi’s glue to a vector bundle C. Since Λij preserves the standard bracket T T −, − S and the standard pairing h−, −i on TgUi, we have a well-defined bracket −, − and a nonde- generateJ K symmetric bilinear form h−, −i on Γ(C). Clearly, we have the following exactJ sequenceK of vector bundles: ρ 0 −→ ker ρ −→ C −→ TM −→ 0, where ρ is induced by the projection TgUi −→ TUi. T T The fact that Λij preserves the standard bracket −, − S and the standard pairing h−, −i also implies that Axioms (i)-(iii) in Definition 4.1 are satisfied. Therefore,J K (C, −, − , h−, −i , ρ) is a transitive Courant algebroid. J K

On Ui, consider the splitting si : TUi −→ C|Ui given by g si(X) = X + θi(X) − (θi, θi(X)) − iX Bi. (39)

T Straightforward calculation shows that hsi(X), si(Y )i = 0. Thus, the splitting si is isotropic. Eqs. (36) and (37) implies that Λijsj(X) = si(X). Thus, we have a globally well-deﬁned isotropic splitting s : TM −→ C. ` Furthermore, Ui × g and the transition function τij give us a Lie algebra bundle G, and there is a short exact sequence ρ0 0 −→ T ∗M −→ ker ρ −→ G −→ 0, 0 ∗ where ρ is induced by the projection (Ui × g) ⊕ T Ui −→ Ui × g. Consider the splitting σsi : G|Ui −→

(ker ρ)|Ui given by g σsi (a) = a − 2(θi, a) . (40) T Then hsi(X), σsi (a)i = 0. Thus, σsi is orthogonal to s. By (36), we have τij 0 a τija τij(a) ? g = ? g = g , −2φijτij 1 −2(θj, a) −2φijτij(a) − 2(θj, a) −2(θi, τij(a))

20 which implies that we have a globally well-deﬁned splitting σs : G −→ ker ρ that orthogonal to the splitting s. Remark 4.5. In Appendix A.3, we write down the explicit formula for the glued transitive Courant algebroid in the form provided in [17].

p A 1-morphism in holim TCc (U(M)•) from (θei, Bei; φeij, τeij, βeij) to (θi,Bi; φij, τij, βij) consists of a 1-  1 0 0  p morphism  φi τi 0  in TCc (∪Ui)1 from t(TgUi, θei, Bei) to t(TgUi, θi,Bi), which satisfies ? βi −2φi τi 1  1 0 0   1 0 0  Λij  φj τ 0  =  φi τ 0  Λeij. (41) ? ? βj −2φj τj 1 βi −2φi τi 1 p Proposition 4.6. A 1-morphism in holim TCc (U(M)•) gives rise to a Courant algebroid isomorphism preserving connections. Proof. The proof is similar to that of Proposition A.4. Eq. (41) implies that the local morphisms glue together to a global morphism B between the gluing results of two Courant data. Eqs. (33) and (34) imply that Bsi(X) = si(X) and Bσ = σs. Thus B preserves connections. e es p+ By Proposition 4.4 and Proposition 4.6, after the plus construction, we arrive at the (2, 1)-sheaf TCc of transitive Courant algebroids with connections, where the 1-morphism are the isomorphisms of Courant algebroid preserving connections. See [29] for the notion of morphisms (not necessary isomorphisms) of Courant algebroids. However, in our case, all our sheaves (stacks) are functors to (higher) groupoids, that is, 1-morphisms are always isomorphisms. p p Obviously, there is a projection pr from the (2, 1)-presheaf TCc to the (2, 1)-presheaf TLc , which T T sends a transitive Courant data t (TgUi, −, − S , h−, −i , prTU , θi,Bi); φij, τij, βij to the transitive J K i T Lie data t (TUi ⊕ (Ui × g), [−, −]S , prTU , θi); φij, τij (see Definition A.2), and behaves in a similar p+ pr p+ obvious way on the level of morphisms. After plus construction, we arrive at a projection TCc −→ TLc . p p+ Here TLc and TLc are the (2, 1)-presheaf and (2, 1)-sheaf of transitive Lie algebroids respectively. See Appendix A.1 for more details. g T Now we fix a transitive Lie data Ac = t (TUi ⊕ (Ui × ), [−, −]S , prTUi , θi); φij, τij . We simply denote it by (θi; φij, τij). 2 Lemma 4.7. Define Ri : ∧ Γ(TUi) −→ Γ(Ui × g) by 1 R = dθ + [θ , θ ] . (42) i i 2 i i g 2 Then Ri’s glue to a globally well-defined curvature R : ∧ TM −→ G. We call R the curvature of our transitive Lie data.

Proof. We need to show τijRj = Ri. We bring (36) inside the expression. Then the result follows from Eqs. (29) and (30). Remark 4.8. It turns out that the curvature form R in this lemma is the same R appearing in the Courant bracket (23) for the gluing result (see Proposition A.9).

Then the ﬁrst Pontryagin class of the transitive Lie data Ac is deﬁned through the curvature R,

g p1(Ac) := (R,R) . (43) p We now describe the condition for the ﬁbre of holim TCc (U(M)•) over a particular Lie algebroid with connection to be non-empty.

21 Ac p+ p+ Theorem 4.9. Given an object M −−→ TLc in TLc over a good cover {Ui} of M,

(i) there exists a lift Cc that ﬁts the diagram

p+ TCc (44) = Cc Ac p+ M / TLc ,

if and only if p1(Ac) = 0;

3 (ii) if a lift exists, then the space of lifts up to isomorphisms forms a torsor of Ωcl(M). The proof of this theorem is given in the following two lemmas.

Lemma 4.10. There exists a lift Cc of Ac in diagram (44) if and only if

p1(Ac) = 0.

T T Proof. First, assume that Cc = t (TgUi, −, − S , h−, −i , prTU , θi,Bi); φij, τij, βij is a lift of Ac, J K i then set Hi := cs3(θi) − dBi on each Ui. By (38), we have cs3(θi) − dBi = cs3(θj) − dBj, which implies that Hi = Hj on Uij. Thus, Hi’s glue to a global 3-form H. We call H the curvature of Cc. Since g 1 g H|Ui = cs3(θi) − dBi, and cs3(θi) = (θi, dθi) + 3 (θi, [θi, θi]) , we have 1 dH| = d((θ , dθ )g + (θ , [θ , θ ])g) = (dθ , dθ )g + (dθ , [θ , θ ])g. Ui i i 3 i i i i i i i i Here we use the fact that (−, −)g is adjoint invariant. Another direct computation gives 1 1 (R ,R )g = (dθ + [θ , θ ] , dθ + [θ , θ ] )g i i i 2 i i g i 2 i i g g i g = (dθi, dθi) + (dθ , [θi, θi]) .

g Therefore, we have (R,R) = dH, i.e. p1(Ac) = 0. g T On the other hand, given Ac = t(TUi⊕(Ui× ), [−, −]S , prTUi , θi); φij, τij and assuming p1(Ac) = 0, let H ∈ Ω3(M) be such that (R,R)g = dH. Since the cover is good, we may come up with 2-forms 2 Bi ∈ Ω (Ui) satisfying

H|Ui = cs3(θi) − dBi. ∗ Let βij : TUij −→ T Uij be the bundle map uniquely determined by (37). Then one checks that T T t (TgUi, −, − S , h−, −i , prTUi , θi,Bi); φij, τij, βij is a Courant data whose image under the projec- p+ Jpr K p+ tion TCc −→ TLc coincides with Ac. g T p+ Now assume that there exists a lift of Ac = t(TUi ⊕(Ui × ), [−, −]S , prTUi , θi); φij, τij ∈ TLc (M),

where again Ac is over a good cover of M. We denote the ﬁber category of pr over Ac by SAc .Then the

space of lifts up to isomorphisms is the set of equivalent classes SAc /1-morphisms. We deﬁne an action 3 3 h of Ωcl(M) on this space as follows. For all h ∈ Ωcl(M), assume that h|Ui = dBi . Deﬁne the action of h on SAc /1-morphisms by

h h h h · [(θi,Bi; φij, τij, βij)] = [(θi,Bi + Bi ; φij, τij, βij + Bj − Bi )]. (45)

Here [] denotes the isomorphism class in the quotient SAc /1-morphisms. Now we prove that the above action is well deﬁned.

22 h ¯h ¯h h • It does not depend on the choice of {Bi }. In fact, if h|Ui = dBi , we let Bi = Bi − Bi , which is  1 0 0  ¯h ¯h ¯h closed. Then  0 1 0  gives rise to an isomorphism from (θi,Bi + Bi ; φij, τij, βij + Bj − Bi ) Bi 0 1 h h h to (θi,Bi + Bi ; φij, τij, βij + Bj − Bi ).

• It does not depend on the choice of a representative (θi,Bi; φij, τij, βij) of an isomorphism class. The isomorphism is possibly given through another ﬁner cover {Vi}. However as we may always pull back our data to Vi, we might as well assume that Vi = Ui. Assume that we have a 1-morphism Λ from a

transitive Courant data (θi, Bei; φij, τij, βeij) ∈ SAc to a transitive Courant data (θi,Bi; φij, τij, βij) ∈ h h h SAc . Then by (35), one checks that Λ is a 1-morphism from (θi, Bei + Bi ; φij, τij, βeij + Bj − Bi ) to h h h (θi,Bi + Bi ; φij, τij, βij + Bj − Bi ).

3 Lemma 4.11. With the above notations, Ωcl(M) acts on SAc /1-morphisms freely and transitively. Thus, 3 SAc /1-morphisms is an Ωcl(M)-torsor.

3 Proof. We ﬁrst show that the action of Ωcl(M) is free. If h · [(θi,Bi; φij, τij, βij)] is isomorphic to h [(θi,Bi; φij, τij, βij)], by (35), we deduce that dBi = 0, which implies that h = 0. 0 0 Then we show that the action is transitive. For two objects (θi,Bi; φij, τij, βij) and (θi,Bi; φij, τij, βij) h 0 0 in SAc , let Bi = Bi − Bi. Since both {cs3(θi) − dBi} and {cs3(θi) − dBi} can be glued to a global 3-form, h we deduce that {dBi } can be glued to a global closed 3-form h. By (37), we deduce that i 0 0 0 g g ? X (Bj − Bi) = βij + (θi, θi(X)) − (θj, θj(X)) − 2φijτij(θj(X)), i g g ? X (Bj − Bi) = βij + (θi, θi(X)) − (θj, θj(X)) − 2φijτij(θj(X)),

which implies that 0 h h βij = βij + Bj − Bi . 0 0 Thus, h · (θi,Bi; φij, τij, βij) = (θi,Bi; φij, τij, βij). It ﬁnishes the proof.

Ac p+ Corollary 4.12. Given an object M −→ TLc with p1(Ac) = 0, the curvatures of all possible lifts form 3 an Ωcl(M)-torsor. p+ Remark 4.13. Note that objects in the category TCc (M) are transitive Courant algebroids with connec- p+ tions and 1-morphisms in TCc (M) are isomorphisms between transitive Courant algebroids that preserve p+ connections. Thus, obviously there is a forgetful functor F from the category TCc (M) to the usual category of transitive Courant algebroids TC(M). Similarly, there is also a forgetful functor from the category p+ TLc (M) to the usual category of transitive Lie algebroids TL(M) over M. Let SA denote the ﬁber of the projection pr : TC(M) −→ TL(M). Then we have the following diagram:

p+ pr p+ SAc / TCc (M) / TLc (M)

F F F pr SA / TC(M) / TL(M).

In [17], the isomorphism classes in SA are classified for a quadratic transitive Lie algebroid A ∈ TL(M) with vanishing first Pontryagin class. Compared to Lemma 4.11, the classification results are not the same 3 since our morphisms need to preserve connections. More precisely, our space is a torsor of Ωcl(M) which surjectively maps to the group H3(M, R)/I of which their space is a torsor. Here I is a certain subspace of H3(M, R) which comes from automorphisms of the Lie algebroid A. In our case, such automorphisms 3 do not show up because our Ωcl(M)-action (45) fixes the underlying transitive Lie data.

23 k Example 4.14 (action Courant algebroids). A quadratic Lie algebra (k, [−, −]k, (−, −) ) gives rise to a string Lie 2-algebra [46, Sect. 2] and may be viewed as a Courant algebroid over a point. Extending this idea to a general base manifold M, the authors in [29] construct a natural example of Courant algebroids as follows: given an action ρ : k → X(M) whose stabilizer at each point on M is a coisotropic subspace of k, there is an action Courant algebroid (M × k, −, − , h−, −i , ρ), where the anchor is given by the action ρ, the bilinear form h−, −i is the pointwise pairingJ K induced by (−, −)k on k, and the Courant bracket on the sections of the vector bundle M × k → M is given by

∗ ∞ X,Y = [X,Y ]k + Lρ(X)Y − Lρ(Y )X + ρ hdX, Y i , ∀X,Y ∈ C (M, k). J K Now we study this example in the special case of homogeneous spaces. Take k = gln(C) equipped k with the nondegenerate bilinear form (A, B) = tr(AB), and let k≥0, k0, and k+ be the Lie subalgebras of non-strict upper triangular, diagonal, and strict upper triangular matrices respectively. Let K = GLn(C), and K≥0, K0, and K+ be the matrix groups corresponding to k≥0, k0, and k+ respectively. Take M to be the homogeneous space K/K≥0. In this case, the anchor ρ is surjective and it follows that the action Courant algebroid M × k is a transitive Courant algebroid. Thus, after choosing a connection, we have a split of Courant algebroid M × k =∼ TM ⊕ G ⊕ T ∗M, with the Courant bracket and the pairing defined in (23) and (24). We now prove that there exists a suitable connection such that the split form on the right hand side is a standard transitive Courant algebroid. Firstly, following [54, Proposition 3.9] the underlying transitive ⊥ × n Lie algebroid A = M × k/(ker ρ) is the Atiyah Lie algebroid associated to the K0 = (C ) principal bundle K/K+ → M. However, this principal bundle is trivial. This can be seen as follows. Note that M n is isomorphic to the flag variety Fn := {E• = (E1 ⊂ E2 ⊂ · · · ⊂ En = C ) | dimEi = i}. Let Ui be the tautological i-dimensional vector bundle over M, whose fiber over a point (a flag E•) is the vector space n Ei of the flag. These bundles form a filtration 0 = U0 ⊂ U1 ⊂ · · · ⊂ Un = Fn × C . If we consider the n standard representation of K0 ⊂ GLn(C) on C , then the associated vector bundle of the K0 principal n bundle K/K+ → M is isomorphic to ⊕i=1Li. Here Li := Ui/Ui−1 is a line bundle for each 0 < i ≤ n. n ∼ Therefore the triviality of K/K+ → M follows from the fact that the associated bundle ⊕i=1Li = Un is trivial. Upon choosing a global trivialization, we can take the natural connection on the principal bundle ∼ K/K+ → M, which in turn induces a split of the Atiyah Lie algebroid A = TM ⊕ (M × k0) with ∗ connection ∇X (a) = X(a). Let TM ⊕ (M × k0) ⊕ T M be the corresponding split of M × k. By Proposition A.9 and the fact that the curvature R of the natural connection is 0, the Courant bracket on the split form is the standard Courant bracket up to a 3-form H ∈ Ω3(M) satisfying dH = (R,R)k0 = 0. Following from Borel’s result, as a complete flag variety, M has vanishing odd cohomology. Thus we may ∗ assume H = dB for a certain 2-form B, and perform a B-field transformation for TM ⊕ (M × k0) ⊕ T M, X + a + ξ 7→ X + a + ξ + iX B, and arrive at the standard bracket. Thus composing these two steps, we obtain an isomorphism from M × k to the standard transitive Courant algebroid. Notice that the K0 principal bundle K/K+ → M is not necessarily trivial for general K. For example, × n−1 7 when K = SLn(C), the similar construction will give us nontrivial K0 = (C ) principal bundle . Since K0 is abelian, the Cartan 3-form on it is 0, thus the basic gerbe on it is a trivial gerbe. Nevertheless, we shall not expect string groups to be trivial. Therefore, there will be different features for abelian counterpart of string structure. We leave it for future discussion.

5 Morphism from the string sheaf to the transitive Courant sheaf

Having constructed the sheaves of String(n)-principal bundles and transitive Courant algebroids with connections, we show that there is a canonical morphism between them. On the level of objects with- out connections, one could build the correspondence between String structures and transitive Courant

7We thank very much Eckhard Meinrenken for sharing this example with us.

24 algebroids using the reduction method as in [5, 13]. Nevertheless, to obtain a functor, it is convenient to use our language. Throughout this section, we take the presheaf of transitive Courant algebroids with connections for G = Spin(n) and g = so(n) with the bilinear form (−, −)g the one appeared in (4). We p still denote this presheaf by TCc .

p+ p+ 5.1 Construction of the morphism Φ: BString(n)c → TCc p p Theorem 5.1. There is a canonical morphism Φ from BString(n)c to TCc , where for any U ∈ Mfd, the p p morphism Φ(U): BString(n)c (U) → TCc (U) is given on 0-, 1- and 2-simplices respectively as follows p • for an object (U × String(n) → U, θ, B) in BString(n)c (U), we have

Φ(U × String(n) → U, θ, B) = (TgU, θ, B);

2 • for a 1-simplex (g01,A01, ω01):(U × String(n) → U, θ1,B1) → (U × String(n) → U, θ0,B0), we have  1 0 0  2 ∗ Φ(g01,A01, ω01) = Λ01 :=  −g¯01θMC adg¯01 0  , ∗ ? β01 2(¯g01θMC) ◦ adg¯01 1

where g¯01 : U → G is the underlying morphism of g01 and

∗ g 2 ∗ ? ∗ β01 = −(¯g01θMC, θ0) + dA01 + ω01 − (¯g01θMC) ◦ g¯01θMC.

• for a 2-simplex (f, ω1), we have Φ(f, ω1) = 1.

∗ ? ∗ Remark 5.2. Note that the symmetric part of β01 is given by −(¯g01θMC) ◦ g¯01θMC, which is the same as the symmetric part of Ψ given in (75) in Appendix A.4. Thus, Λ01 is an inner automorphism in the sense of Sˇevera [42]. To prove this theorem, we need the following lemmas. Lemma 5.3. For any G-valued function g : U −→ G, and a, b ∈ Γ(U × g), X,Y ∈ Γ(TU), we have

[adga, adgb]g = adg[a, b]g, (46) ∗ ∗ ∗ ∗ ∗ X(g θMC(Y )) − Y (g θMC(X)) − [g θMC(X), g θMC(Y )]g = g θMC([X,Y ]), (47) ∗ X(adgb) − [g θMC(X), adgb]g = adg(X(b)). (48)

1 Proof. (46) is obvious. (47) follows from the Maurer-Cartan equation dθMC − 2 [θMC, θMC]g = 0. For any d m ∈ U, let γ(s) be the integration curve of X through m, i.e. Xm = ds |s=0γ(s). Then we have d X (ad b) = X ( | Ad exp(tb)) m g m dt t=0 g d d = | | Ad exp(tb(γ(s))) dt t=0 ds s=0 g(γ(s)) d d = | Ad exp(tX (b)) + | Ad exp(tb(m)) dt t=0 g(m) m ds s=0 g(γ(s)) d = ad X (b) + | ad −1 ad b(m) g(m) m ds s=0 g(γ(s))·g(m) g(m) ∗ = adg(m)Xm(b) + [g(m) θMC(Xm), adg(m)b(m)]g, which implies that (48) holds. The proof is ﬁnished.

2 Lemma 5.4. Let (g01,A01, ω01) be a 1-morphism from (U × String(n) → U, θ1,B1) to (U × String(n) → p U, θ0,B0). Then Λ01 given in Theorem 5.1 is a 1-morphism in TCc from (TgU, θ1,B1) to (TgU, θ0,B0).

25 Proof. By deﬁnition, we ﬁrst need to show that Λ01 is indeed an automorphism of the standard transitive T T Courant algebroid (TgU, −, − S , h−, −i , prTU ). That is to prove the entries of the vector bundle map  1J 0K 0  ∗ Λ01 =  −g¯01θMC adg¯01 0  satisfy the identities (28)-(31). Note that (g01 : U → String(n), ∗ g β01 2(¯g01θMC, ·) 1 1 2 2 A01 ∈ Ω (U), ω01 ∈ Ω (U)) gives rise to a 1-morphism,

2 (g01,A01,ω01) (U × String(n) → U, θ0,B0) ←−−−−−−−− (U × String(n) → U, θ1,B1),

if and only if

2 2 ∗ dA01 = ω01 + B1 − B0, dω01 = cs3(θ1) − cs3(θ0), θ0 − adg¯01 θ1 = −g¯ θMC, (49)

where g¯01 : U → G is the underlying morphism of g01 : U → String(n). ∗ ∗ ? ∗ sym The symmetric part of β01 : TU −→ T U is −(¯g01θMC) ◦g¯01θMC, which we denote by β01 . Therefore, 1 β (X)(Y ) + β (Y )(X) = βsym(X)(Y ) = −(−g¯∗ θ (X), −g¯∗ θ (Y ))g, 2 01 01 01 01 MC 01 MC which implies that (28) holds. By (47) and (48), we deduce that (29) and (30) hold. ∗ g 2 skew The skewsymmetric part of β01 is −(¯g01θMC, θi) + dA01 + ω01, which we denote by β01 . Obviously, we have skew i skew skew skew hLX β01 (Y ) − Y dβ01 (X) − β01 ([X,Y ]),Zi = dβ01 (X,Y,Z). Furthermore, we have 1 cs (θ ) − cs (θ ) = d(θ , g∗ θ )g − g¯∗ C, (50) 3 1 3 0 0 01 MC 6 01 where C ∈ Ω3(G) defined by 1 C(â, ˆb, cˆ) = (θ (â), [θ (ˆb), θ (ˆc)] )g = (a, [b, c] )g, ∀a, b, c ∈ g. 6 MC MC MC g g

Here a,ˆ ˆb, cˆ are right invariant vector ﬁelds on G. By (49) and (50), we obtain 1 dβskew = cs (θ ) − cs (θ ) − d(¯g∗ θ , θ )g = − g¯∗ C. (51) 01 3 1 3 0 01 MC i 6 01 On the other hand, by straightforward computations, we have

sym i sym sym ∗ ∗ ∗ hLX β01 (Y ) − Y dβ01 (X) − β01 ([X,Y ]) + P(¯g01θMC, g¯01θMC),Zi =g ¯01C(X,Y,Z), which implies that (31) holds. p Finally, (49) implies that the conditions in (33)-(35) are satisﬁed. Thus Λ01 is a 1-morphism in TCc . It ﬁnishes the proof.

Recall that the model we use for a 2-groupoid is a simplicial set satisfying Kan conditions Kan(n, j) for all n ≥ 1 and 0 ≤ j ≤ n and strict Kan conditions Kan!(n, j) for all n ≥ 3 and 0 ≤ j ≤ n. Lemma p p 5.4 veriﬁes that Φ(U): BString(n)c (U)1 → TCc (U)1 is well-deﬁned. The following lemma will verify that p p the map Φ(U): BString(n)c (U)2 → TCc (U)2 on 2-simplices commutes with the face maps. 1 2 Lemma 5.5. Given any 2-morphism (f, ω ) between T01◦T12 and T02, where Tij := (¯gij,Aij, ωij){0≤i

26 2 the corresponding images Λij := Φ(Tij) = Φ(¯gij,Aij, ωij){0≤i

dition Λ01Λ12 = Λ02, i.e. we have the following commutative diagram:

(θ1,B1)

Λ Λ 01 Id 12

Λ02 (θ0,B0) (θ2,B2).

Proof. By g¯01g¯12 =g ¯02, we have

 1 0 0   1 0 0  ∗ ∗ Λ01Λ12 =  −g¯01θMC adg¯01 0   −g¯12θMC adg¯12 0  ∗ ? ∗ ? β01 2(¯g01θMC) ◦ adg¯01 1 β12 2(¯g12θMC) ◦ adg¯12 1  1 0 0  ∗ ∗ =  −g¯01θMC − adg¯01 g¯12θMC adg¯01 adg¯12 0  ,

D31 D32 1 where

∗ ? ∗ ? D32 = 2(¯g01θMC) ◦ adg¯01 ◦ adg¯12 + 2(¯g12θMC) ◦ adg¯12 ∗ ∗ ? = 2(¯g01θMC + adg¯01 g¯12θMC) ◦ adg¯02 ∗ ? = 2(¯g02θMC) ◦ adg¯02 , and

∗ ? ∗ D31 = β01 − 2(¯g01θMC) ◦ adg¯01 ◦ g¯12θMC + β12 ∗ g 2 ∗ ? ∗ ∗ ? ∗ = −(¯g01θMC, θ0) + dA01 + ω01 − (¯g01θMC) ◦ g¯01θMC − 2(¯g01θMC) ◦ adg¯01 ◦ g¯12θMC ∗ g 2 ∗ ? ∗ −(¯g12θMC, θ1) + dA12 + ω12 − (¯g12θMC) ◦ g¯12θMC ∗ ∗ g 2 ∗ ? ∗ ∗ = −(¯g01θMC + adg¯01 g¯12θMC, θ0) + dA02 + ω02 − (¯g01θMC) ◦ (¯g01θMC + adg¯01 ◦ g¯12θMC) ∗ ? ∗ ∗ ? ∗ −(adg¯01 ◦ g¯12θMC) ◦ g¯01θMC − (adg¯01 ◦ g¯12θMC) ◦ adg¯01 ◦ g¯12θMC ∗ g 2 ∗ ? ∗ ∗ ? ∗ = −(¯g02θMC, θ0) + dA02 + ω02 − (¯g01θMC) ◦ g¯02θMC − (adg¯01 ◦ g¯12θMC) ◦ g¯02θMC

= β02.

Therefore, we have Λ01Λ12 = Λ02.

Proof of Theorem 5.1. Recall that given any manifold U, a morphism between the 2-groupoids p p BString(n)c (U) and TCc (U) is a simplicial morphism of the underlying simplicial sets. Lemma 5.4 and p p 5.5 verify that Φ(U): BString(n)c (U) → TCc (U) is indeed a morphism of the underlying simplicial sets. Hence we only need to prove the naturality of the map Φ. To do this, let us assume that rV,U : V → U is a smooth map between two manifolds V and U. Then rV,U induces maps

p p p • TCc (rV,U ): TCc (U) → TCc (V ) given by

∗ ∗ (TgU, θ, B) 7→ (TgV , rV,U (θ), rV,U (B)),

p p p • BString(n)c (rV,U ): BString(n)c (U) → BString(n)c (V ) given by

2 1 ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ 1 ((U, θ, B); g, A, ω ; f, ω ) 7→ (V, rV,U (θ), rV,U (B)); rV,U (g), rV,U (A), rV,U (ω ); rV,U (f), rV,U (ω )).

27 p Here (U, θ, B) stands for an element (U × String(n) → U, θ, B) ∈ BString(n)c (U)1. It is then straightforward to obtain the following commutative diagram

p Φ(U) p BString(n)c (U) −−−−→ TCc (U)   BString(n)p(r ) TCp(r ) c V,U y c V,U y p Φ(V ) p BString(n)c (V ) −−−−→ TCc (V ),

p p which shows the naturality of the map Φ. Thus Φ is morphism from BString(n)c to TCc . As a corollary, we have proven the desired result,

p+ Corollary 5.6. There is a natural morphism Φ from the (3, 1)-sheaf BString(n)c to the (2, 1)-sheaf p+ TCc . By the discussion above, the morphism Φ can be described explicitly as follows. Given any manifold p+ p+ M, the morphism Φ: BString(n)c (M) → TCc (M) is given on the 0-, 1- and 2-simplices respectively by

• on 0-simplices ∗ Φ({Ui},Pc) = {Ui}, (t(TgUi, θi,Bi); −g¯ijθMC, adg¯ij , βij)

2 1 where {Ui} is an open cover of M and Pc := (t(Ui × String(n) → Ui, θi,Bi; gij,Aij, ωij; fijk, ωijk) is string data, g¯ij : Uij → G is the underlying morphism of gij, and βij is given by

∗ g 2 ∗ ? ∗ βij = −(¯gijθMC, θi) + dAij + ωij − (¯gijθMC) ◦ g¯ijθMC.

• on 1-simplices Φ({Vi}, φc) = (TgVi, Λi), 2 where {Vi} is a common reﬁnement of {Ui}, {Uei}, and φc := (gi : Vi → String(n),Ai, ωi ) provides a 1-morphism between ({Ui},Pc) and ({Uei}, Pec), Λi is the 1-morphism from (TgVi, θei, Bei) to (TgVi, θi,Bi) deﬁned as before  1 0 0  ∗ Λi =  −g¯i θMC adg¯i 0  , ∗ ? βi 2(¯gi θMC) 1

∗ where g¯i : Vi → G is the underlying morphism of gi and βi : TVi −→ T Vi is given by

∗ g 2 ∗ ? ∗ βi = −(¯gi θMC, θi) + dAi + ωi − (¯gi θMC) ◦ g¯i θMC;

• on 2-simplices Φ({Wi}, αc) = 1,

where {Wi} is a common reﬁnement of {Vi}, {Vei}, and αc provides a 2-morphism between ({Vi}, φc) and ({Vei}, φec). p+ As an object in TCc (M) glues to a Courant algebroid by the discussion in Appendix A.3, let us describe explicitly the Courant algebroid with a connection associated to a String(n)-principal bundle with connection data on a manifold M. Given a String(n) data

2 1 Pc = (tUi × String(n) → tUi, θi,Bi; gij,Aij, ωij; fijk, ωijk)

28 p over a cover {Ui} of M in BString(n)c , the corresponding transitive Courant algebroid with a connection p+ p + under the morphism Φ: BString(n)c (M) → TCc (M) is

∼ ∗ T T E = (TM ⊕ G ⊕ T M, −, − ∇,R,H , h−, −i , prTM ), J K T T equipped with bracket −, − ∇,R,H and pairing h−, −i as in formula (23), (24) with the global 2-form R ∈ Ω2(M, G), the connectionJ K ∇ : Γ(TM) ⊗ Γ(G) −→ Γ(G) and the global 3-form H ∈ Ω3(M) deﬁned on each Ui respectively by

i ∇X a := X(a) + [θi(X), a]g, ∀X ∈ Γ(TUi), a ∈ Γ(Ui × g), 1 R := dθ + [θ , θ ] , i i 2 i i g Hi := cs3(θi) − dBi.

p+ p+ 5.2 Property of the morphism Φ: BString(n)c → TCc We denote the (3, 1)-presheaf of U(1)-gerbes (or BU(1)-principal bundles) with connection data by p ι p Bι BBU(1)c . Then, induced by the morphism BU(1) −→ String(n), there is a morphism BBU(1)c −→ p BString(n)c given by

(U × BU(1),B ∈ Ω2(U); L : U → BU(1),A ∈ Ω1(U); a : U → U(1)) 7→ (U × String(n), B, θ = 0; g = ι ◦ L, A, ω2 = 0; a, ω1 = 0).

p On the side of algebroids, we have similar constructions. Let us denote by ECc the (2, 1)-presheaf of exact Courant algebroids (see Appendix A.2 for the deﬁnition and notations). There is a morphism on p p the presheaf level, ECc → TCc , given by 1 0 0 1 0 ( U, −, − E , h−, −iE, pr ,B; ) 7→ ( U, −, − T , h−, −iT , pr , 0,B; 0 1 0 ). T S TU B 1 Tg S TU   J K J K B 0 1

p Let us denote by TCc the (2, 1)-presheaf of transitive Lie algebroids (see Appendix A.1). Similar to the p π p p p morphism BString(n)c −→ BSpin(n)c described before Theorem 3.2, we have a morphism TCc → TLc deﬁned also by forgetting some data,

1 0 0 1 0 ( U, −, − T , h−, −iT , pr ; θ, B; φ τ 0 ) 7→ (TU ⊕ (U × g), [−, −]T , pr , θ; ). Tg S TU   S TU φ τ J K β −2φ? ◦ τ 1

p+ p+ Then we can also construct morphisms ΦU(1) : BBU(1)c → ECc and ΦSpin(n) : BSpin(n)c → TLc , for g = so(n). The constructions are essentially given in [42, 31], and here we spell it out in our setting. On the level of objects, they are given by

∗ 1 0 ΦU(1) :(Ui × BU(1),Bi; Lij,Aij; aijk) 7→ (TUi ⊕ T Ui,Bi; ), dAij 1 (52) T 1 0 ΦSpin(n) :(Ui × Spin(n), θi; gij) 7→ (TUi ⊕ (Ui × so(n)), [−, −]S , prTU , θi; ∗ ), −gijθMC adgij and on the level of morphisms, they are given by corresponding pullbacks and pre-compositions.

29 Thus, we have the following commutative diagram to connect the principal bundle side and the algebroid side, Φ U(1) p+ BBU(1)c / ECc (53)

Bι p+ Φ p+ BString(n)c / TCc

π Φ Spin(n) p+ BSpin(n)c / TLc .

Now we show that ΦU(1) is not injective in general and this implies that Φ is not injective. And ΦSpin(n) is not surjective thus Φ can not be surjective. Moreover, even on the ﬁbre of the image of ΦSpin(n), Φ can not be surjective in general.

3 Lemma 5.7. When H (M, Z) has torsion, ΦU(1) is not injective on the level of objects and not fully faithful.

0 3 Proof. We take a cocycle aijk representing a torsion element in H (M, Z), lifting it to a Deligne cocycle 0 0 0 0 (aijk,Aij,Bi ), then dBi glues to an exact 3-form. We now show that such a Deligne cocycle may always 0 0 0 0 0 0 be adjusted by an exact one to (aijk,Aij,B |Ui ) for a global 2-form B and some closed Aij. Since dBi 0 0 0 glues to an exact 3-form, there is a global 2-form B such that Bi = B|Ui + dAi . Then adjusting the 0 original Deligne cocycle by D(1,Ai ) will fulﬁll our aim. 0 0 0 0 Therefore we might as well assume that we lift aijk to a Deligne cocycle (aijk,Aij,B |Ui) satisfying 0 dAij = 0. Then clearly the image of the two objects (Ui × BU(1),Bi; Lij,Aij; aijk) ∈ BBU(1) and 0 0 0 (Ui × BU(1),Bi + Bi ; Lij,Aij + Aij; aijk + aijk) ∈ BBU(1) are the same. However, it is clear that there 0 exists no morphism between these two objects because aijk is not exact. Lemma 5.8. In general, the map Φ is not injective on the level of objects and not fully faithful .

Proof. We take the two diﬀerent gerbes with connection data constructed in Lemma 5.7, G1 and G2, which maps to the same object under ΦU(1). Then we see that Bι(G1) and Bι(G2) are non-isomorphic string data but mapping to the same Courant data on the right hand side. Lemma 5.9. The map Φ preserves curvatures. Proof. It is clear from the deﬁnition of curvatures on both sides.

Lemma 5.10. The map Φ is not essentially surjective in general8.

Proof. The map ΦSpin(n) is not essentially surjective because there are non-integrable transitive Lie algebroids. To show Φ is not essentially surjective, we need to ﬁnd a non-integrable transitive Lie algebroid A whose p1(A) = 0. Notice that integrability is a property preserved by isomorphisms of Lie algebroids. 3 3 We take M = R − {p1} − {p2} where p1, p2 ∈ R are two diﬀerent points and A = TM × R, with the following Lie bracket [(X, f), (Y, g)] = [X,Y ] + X(f) − Y (g) + ω(X,Y ). √ ∗ ∗ ∗ ∗ 3 3 2 Here ω = ι1pr1ωa + 2ι2pr2ωa with ιj : M → R − pj, prj : R − pj → S for j = 1, 2, and ωa the area 2 R form on S . Then the period { γ ω, γ ∈ π2(M)} of ω is dense in R. Therefore, A is not integrable by [19]. On the other hand, it is clear p1(A) = 0 because there is no non-trivial 4-forms on M.

Lemma 5.11. On the ﬁbre of the image of ΦSpin(n), Φ is in general not essentially surjective.

8We thank very much Pavol Ševera for pointing out this example to the third author.

30 P¯c Proof. As pointed out in Lemma 5.9, Φ preserves the curvature. If we fix an object M −→ BSpin(n)c and ¯ look at all possible lifts of P¯c, we see that curvatures for these lifts form a torsor of im d as in Corollary 3.8. Now on the Courant side, fixing the underlying Spin(n)-principal bundle and its connection data means that we consider all possible Courant lifts over a fixed Atiyah Lie algebroid and its connection 3 data. By Corollary 4.12, the set of curvatures for such Courant lifts is a torsor of Ωcl(M). As pointed out ¯ 2 d 3 in Remark 3.9, H (M,D2) −→ Ωcl(M) is not surjective in general. Since curvatures are preserved under isomorphisms, we see that Φ can not be essentially surjective in general even on the fibre of the image of ΦSpin(n).

A Appendix

p+ A.1 (2, 1)-sheaf TLc of transitive Lie algebroids with connections Deﬁnition A.1. A Lie algebroid structure on a vector bundle A −→ M is a pair that consists of a Lie algebra structure [−, −] on the section space Γ(A) and a bundle map ρ : A −→ TM, called the anchor, such that the following relation is satisﬁed:

[X, fY ] = f[X,Y ] + ρ(X)(f)Y, ∀ X,Y ∈ Γ(A), f ∈ C∞(M).

A Lie algebroid A is called transitive if ρ is surjective, i.e. im ρ = TM. Denote by G = ker ρ. Then G is a bundle of Lie algebras, whose ﬁbre is isomorphic to a Lie algebra (g, [−, −]g). We have the following short exact sequence: ρ 0 −→ G −→ A −→ TM −→ 0. A splitting s : TM −→ A gives rise to a connection ∇ on G by

∇X a = [s(X), a], ∀ X ∈ Γ(A), a ∈ Γ(G).

Thus we call such a splitting s : TM −→ A a connection of A. Connections always exist by partition of unity. Thus, after picking a connection, we have A =∼ TM ⊕ G, and the induced bracket on TM ⊕ G is

T [X + a, Y + b]∇ = [X,Y ] + ∇X b − ∇Y a + [a, b]g + R(X,Y ), ∀ X,Y ∈ Γ(TM), a, b ∈ Γ(G), (54) where R(X,Y ) = [s(X), s(Y )] − s([X,Y ]) is the curvature of the connection s. In other words, a T transitive Lie algebroid with a connection is always isomorphic to (TM ⊕ G, [·, ·]∇, ρ = prTM ) and the isomorphism depends on the choice of the connection. In particular, if G = M × g is a trivial bundle and the connection ∇ is given by the ﬂat connection ∇X b = X(b), we obtain the standard bracket

T [X + a, Y + b]S = [X,Y ] + X(b) − Y (a) + [a, b]g. (55)

1 0 An automorphism of the standard transitive Lie algebroid is given by a matrix , where φ τ τ : M −→ Aut(g) and φ ∈ Ω1(M, g) satisfy

φ([X,Y ]) = X(φ(Y )) − Y (φ(X)) + [φ(X), φ(Y )]g,

τ([a, b]g) = [τ(a), τ(b)]g,

τ(X(b)) = X(τ(b)) + [φ(X), τ(b)]g.

p op There is a (2, 1)-presheaf of transitive Lie algebroids with connections TLc : Mfd → Gpd, where Mfdop is the opposite category of Mfd, and Gpd is the 2-category of (discrete) groupoids and groupoid morphisms. p For an object U ∈ Mfd, the groupoid TLc (U) is made up by the following data:

31 p T 1 • TLc (U)0: an object is a quadruple (TU ⊕(U ×g), [−, −]S , prTU , θ), where θ ∈ Ω (U, g) is a g-valued T 1-form and [−, −]S is the standard bracket given by (55). We will simply denote an object by (TU ⊕ (U × g), θ).

p • TLc (U)1: a 1-morphism from (TU ⊕ (U × g), θe) to (TU ⊕ (U × g), θ) is an automorphism of the 1 0 standard transitive Lie algebroid (TU ⊕ (U × g), [−, −]T , pr ) given by the matrix , S TU φ τ such that θ(X) − τ(θe(X)) = φ(X). The composition of 1-morphisms is simply the matrix multiplication.

p p p Then for a morphism ϕ : U → V in Mfd, the associated functor TLc (ϕ): TLc (V ) → TLc (U) is induced by pulling back forms. p Take an open cover {Ui} of M ∈ Mfd. An object in holim TLc (U(M)•) consists of p • an object t(TUi ⊕ (Ui × g), θi) in TLc (tUi)0, 1 0 p • Λij = ∈ TLc (tUij)1, which is a 1-morphism from (TUij ⊕(Uij ×g), θj|Uij ) to (TUij ⊕ φij τij

(Uij × g), θi|Uij ), therefore satisfying

φij([X,Y ]) = X(φij(Y )) − Y (φij(X)) + [φij(X), φij(Y )]g, (56)

τij([a, b]g) = [τij(a), τij(b)]g, (57)

τij(X(b)) = X(τij(b)) + [φij(X), τij(b)]g. (58)

θi|Uij − τijθj|Uij = φij. (59)

• compatibility condition ΛijΛjk = Λik on Uijk, which unpacks itself to the following two equations

φij + τijφjk = φik,

τijτjk = τik.

p Definition A.2. We call an object t (TUi ⊕ (Ui × g), θi), φij, τij in holim TLc (U(M)•) a transitive Lie data. 1 0 Given a transitive Lie data t (TUi ⊕ (Ui × g), θi), φij, τij , since Λij = satisfies the φij τij cocycle condition ΛijΛjk = Λik, we can glue TUi ⊕ (Ui × g)’s and obtain a vector bundle a A = TUi ⊕ (Ui × g)/ ∼, (60) where the equivalence relation ∼ is given by Y X X + a Y + b ⇐⇒ = Λ , ∀ X + a ∈ TU ⊕ (U × g),Y + b ∈ TU ⊕ (U × g). ∼ b ij a j j i i Proposition A.3. A transitive Lie data t (TUi ⊕ (Ui × g), θi), φij, τij gives rise to a transitive Lie algebroid (A, [−, −], ρ) with a connection s : TM −→ A. Proof. Obviously, the vector bundle A fits the following short exact sequence:

ρ 0 −→ G −→ A −→ TM −→ 0, where G denotes the Lie algebra bundle obtained from the transition function τij and ρ is induced by the projection TUi ⊕ (Ui × g) −→ TUi.

32 T Since Λij preserves the Lie bracket [−, −]S , there is a well-deﬁned Lie bracket [−, −] on Γ(A). Then we obtain a Lie algebroid (A, [−, −], ρ).

On Ui, consider the splitting si : TUi −→ A|Ui given by

si(X) = X + θi(X).

By (59), we have Λijsj(X) = si(X), which implies that we have a global splitting s : TM −→ A.

T T A 1-morphism from t(TUi⊕(Ui×g), [−, −]S , θei), φeij, τeij to t(TUi⊕(Ui×g), [−, −]S , θi), φij, τij in p 1 0 holim TLc (U(M)•) consists of a 1-morphism from t(TUi ⊕(Ui ×g), θei) to t(TUi ⊕(Ui ×g), θi) φi τi p in TLc (∪Ui)1, which satisﬁes 1 0 1 0 Λij = Λeij. (61) φj τj φi τi We have

p Proposition A.4. A 1-morphism in holim TLc (U(M)•) gives rise to a Lie algebroid isomorphism preserving connections.

Proof. Denote by (A,e [−, −e], ρ) (respectively (A, [−, −], ρ)) the transitive Lie algebroid with the connec- e tion se : TM −→ Ae (respectively s : TM −→ A) obtained from the object t (TUi ⊕ (Ui × g), θei), φeij, τeij p (respectively t (TUi ⊕ (Ui × g), θi), φij, τij ) in holim TLc (U(M)•). Thanks to (61), a 1-morphism 1 0 T { } from t (TUi ⊕ (Ui × g), θei), φeij, τij to t (TUi ⊕ (Ui × g), [−, −]S , θi), φij, τij glues to φi τi e a bundle map which gives rise to a Lie algebroid isomorphism between (A,e [−, −e], ρe) and (A, [−, −], ρ). Furthermore, we have

1 0 si(X) = X + φi(X) + τiθei(X) = X + θi(X) = si(X), φi τi e which implies that the connections are also preserved. Remark A.5. The above way to glue a transitive Lie algebroid is essentially the same as the one given by Mackenzie [31]. By Proposition A.3 and A.4, it is not hard to see that after the plus construction we arrive at a p+ (2, 1)-sheaf TLc which maps to the category of transitive Lie algebroids with connections essentially surjectively and fully faithfully.

p+ A.2 (2, 1)-sheaf ECc of exact Courant algebroids with connections ∗ E E E The standard Courant algebroid is (TM ⊕ T M, −, − S , h−, −i , prTM ), where −, − S is the standard Dorfman bracket given by J K J K E i X + ξ, Y + η S = [X,Y ] + LX η − Y dξ, (62) J K and h−, −iE is the canonical symmetric bilinear form given by 1 hX + ξ, Y + ηiE = ξ(Y ) + η(X), (63) 2 A Courant algebroid C is called exact if we have the following short exact sequence

ρ∗ ρ 0 −→ T ∗M −→ C −→ TM −→ 0.

33 A connection of an exact Courant algebroid C is an isotropic splitting9 s : TM −→ C. As before, connections always exist. By choosing a connection s : TM −→ C, the vector bundle C is isomorphic to TM ⊕T ∗M. Then transferring the Courant algebroid structure on C to that on TM ⊕T ∗M, we obtain the ∗ E Courant algebroid (TM ⊕ T M, −, − h , h−, −i , prTM ), where the nondegenerate symmetric pairing E J K E h−, −i is given by (63) and the bracket −, − h is given by J K E E i i X + ξ, Y + η h = X + ξ, Y + η S + Y X h. (64) J K J K 3 Here h ∈ Ωcl(M), defined by h(X,Y ) = s(X), s(Y ) − s[X,Y ], is the curvature of the connection s. In [45], the authors show that exact CourantJ algebroidsK over M are classified by H3(M, R). Now we construct the (2, 1)-presheaf of exact Courant algebroids with connections over the category of (differential) manifolds Mfd. For simplicity, for an object U ∈ Mfd, we write TU := TU ⊕ T ∗U. p op There is a (2, 1)-presheaf of exact Courant algebroids with connections ECc : Mfd → Gpd, where Mfdop is the opposite category of Mfd, and Gpd is the 2-category of (set theoretical) groupoids and groupoid morphisms. p For an object U ∈ Mfd, the groupoid ECc (U) is made up by the following data:

p E E 2 • ECc (U)0: an object is a quintuple (TU, −, − S , h−, −i , prTU ,B), where B ∈ Ω (U) is a 2-form, E E J K −, − S and h−, −i are given by (62) and (63) respectively. We will simply denote an object by (JTU,BK ). p • ECc (U)1: a 1-morphism from (TU, Be) to (TU,B) is a bundle automorphism of TU given by the 1 0 matrix , where B ∈ Ω2(U) is a closed 2-form such that B −B = B. This matrix preserves B 1 e E E the standard Courant bracket −, − S and the pairing h−, −i . The composition of 1-morphisms is simply the matrix multiplication.J K Then for a morphism ϕ : U → V in Mfd, as in the case of Lie algebroids, the associated functor p p p ECc (ϕ): ECc (V ) → ECc (U) is induced by pulling back forms. p Take an open cover {Ui} of M ∈ Mfd. An object in holim ECc (U(M)•) consists of p • an object t(TUi,Bi) in ECc (tUi)0, 1 0 p • Λij = ∈ ECc (tUij)1 which is a 1-morphism from (TUij,Bj|Uij ) to (TUij,Bi|Uij ), Bij 1 therefore satisfying

Bj|Uij − Bi|Uij = Bij.

• compatibility conditions ΛijΛjk = Λik on Uijk which automatically holds.

p+ p+ The plus construction gives us a (2, 1)-sheaf ECc . For a manifold M, an object of ECc (M) consists p of a cover {Ui} and a U(M)-equivariant object of ECc described above. Naturally we ask what the above data glues to. It turns out that the gluing result is an exact Courant algebroid with a connection. The gluing procedure is the same as the one given in [35]. To be self-contained, we give the result using the language of this paper.

p Proposition A.6. An object t (TUi,Bi), Bij in holim ECc (U(M)•) gives rise to an exact Courant algebroid (C, −, − , h−, −i , ρ) with a connection s : TM −→ C. J K 9A splitting s : TM −→ C is called isotropic if the image of s is an isotropic subbundle, i.e. hs(X), s(Y )i = 0, for all X,Y ∈ Γ(TM).

34 p Proof. Given an object t(TUi,Bi), Bij in holim ECc (U(M)•), as before, the cocycle condition ΛijΛjk = Λik implies that TUi’s glue to a vector bundle C via transition matrices Λij’s. Since Λij preserves the E standard bracket −, − S , we have a well-deﬁned bracket −, − on Γ(C). Furthermore, since Λij also J K E J K preserves the standard pairing h−, −i on TUi, we obtain a global nondegenerate symmetric bilinear form h−, −i on C. Obviously, C ﬁts the following exact sequence of vector bundles,

ρ∗ ρ 0 −→ T ∗M −→ C −→ TM −→ 0, where ρ is induced by the projection TUi −→ TUi. E E Also, by the facts that Λij preserves the standard bracket −, − S and the standard pairing h−, −i , Axioms (i)-(iii) in Deﬁnition 4.1 are satisﬁed. Therefore, (JC, −K, − , h−, −i , ρ) is an exact Courant algebroid. J K The 2-forms {Bi} induce an isotropic splitting s : TM −→ C via

s(X) = X − iX Bi,X ∈ Ui.

Note that the deﬁnition of s does not depend on choices of Ui. In fact, if X ∈ Ui ∩Uj, it is straightforward i i to see that X − X Bi ∼ X − X Bj.

p In holim ECc (U(M)•), a 1-morphism from an object t(TUi, Bei), Beij to another object t(TUiBi), Bij 1 0 p consists of a 1-morphism from t(TUi, Bei) to t(TUi,Bi) in ECc (∪Ui)1, which satisﬁes Bi 1 1 0 1 0 Λij = Λeij. (65) Bj 1 Bi 1 Then we have p Proposition A.7. A 1-morphism in holim ECc (U(M)•) gives rise to an exact Courant algebroid isomorphism preserving connections. Proof. The proof is similar to that of Proposition A.4. Eq. (65) is the important information which implies the gluing result. The fact that the bundle map B also preserves the connection, namely B(se(X)) = s(X), follows from the following calculation, i i B(se(X)) = X − X Bei + Bi = X − X Bi = s(X), ∀X ∈ Γ(TM). The proof is ﬁnished.

Similar to the case of transitive Lie algebroids, after the plus construction, we arrive at the (2, 1)-sheaf p+ ECc of exact Courant algebroids with connections.

A.3 Gluing transitive Courant algebroids via local data In this subsection, we give the explicit formula for the transitive Courant algebroid glued by pieces of standard transitive Courant algebroids in a transitive Courant data. This also shows how we may obtain the bracket of a general transitive Courant algebroid from the standard one. As in Proposition 4.4, given a transitive Courant data Cc, there is a corresponding transitive Courant algebroid (C, −, − , h−, −i , ρ). ∗ Using the two splittings s and σs given in (39) and (40), we obtain an isomorphism S : TMJ ⊕G⊕K T M −→ C given by S(X + a + ξ) = s(X) + σs(a) + ξ. (66) ∗ Recall that locally, Si = S|Ui : TUi × (Ui × g) × T Ui −→ C|Ui and its inverse are given by g g Si(X + a + ξ) = X + θi(X) − (θi, θi(X)) − iX Bi + a − 2(θi, a) + ξ, −1 g i g Si (X + a + ξ) = X − θi(X) − (θi, θi(X)) + X Bi + a + 2(θi, a) + ξ.

35 Having TM ⊕ G ⊕ T ∗M equipped with the pairing given by (24), a straightforward computation shows that S preserves the pairing.

i Lemma A.8. Deﬁne ∇ : Γ(TUi) ⊗ Γ(Ui × g) −→ Γ(Ui × g) by

i ∇X a = X(a) + [θi(X), a]g, ∀X ∈ Γ(TUi), a ∈ Γ(Ui × g). (67)

Then, we have j i τij∇X a = ∇X τija, ∀X ∈ Γ(TUij), a ∈ Γ(Uij × g). Thus, by gluing ∇i, we obtain a globally well-deﬁned connection ∇ : Γ(TM) ⊗ Γ(G) −→ Γ(G).

Proof. By (29) and (36), we have

j i τij∇X a − ∇X τija = τij(X(a) + [θj(X), a]g) − X(τija) − [θi(X), τija]g = τij(X(a)) + [τijθj(X), τija]g) − X(τija) − [τijθj(X) + φij(X), τija]g = 0.

The proof is ﬁnished.

Now we see that given a Courant data Cc, we have a connection ∇, a curvature R of the underlining ∗ Lie data Ac given in Lemma 4.7 and a curvature H of Cc. Thus TM ⊕ G ⊕ T M may be equipped with T a transitive Courant algebroid structure with the Courant bracket −, − ∇,R,H given as in (23). J K Proposition A.9. The morphism S in (66) is an isomorphism of Courant algebroids.

∗ Proof. We pull back the bracket on Γ(C) to Γ(TM ⊕ G ⊕ T M) via S and denote it by −, − ind. The T J K only nontrivial thing to check is that −, − ind = −, − ∇,R,H . For all a, b ∈ Γ(Ui × g), we have J K J K −1 T −1 g g T −1 a, b ind = Si Si(a), Si(b) S = Si a − 2(θi, a) , b − 2(θi, b) S = Si ([a, b]g + P(a, b)) J K J K J g K = [a, b]g + P(a, b) + 2(θi, [a, b]g) .

By (67) and Lemma A.8, we have

g g g P(a, b) + 2(θi, [a, b]g) (Y ) = 2(b, Y (a)) + 2(θi(Y ), [a, b]g) g i g = 2(b, Y (a) + [θi(Y ), a]g) = 2(b, ∇Y a) = P (a, b)(Y ),

which implies that

a, b ind = [a, b]g + P (a, b). (68) J K For all X ∈ Γ(TUi), b ∈ Γ(Ui × g), we have

−1 T −1 g i g T X, b ind = Si Si(X), Si(b) S = Si X + θi(X) − (θi, θi(X)) − X Bi, b − 2(θi, b) S J K −1 J K g J K = Si (X(b) − 2LX (θi, b) + [θi(X), b]g + P(θi(X), b)) g g = X(b) + [θi(X), b]g − 2LX (θi, b) + P(θi(X), b) + 2(θi,X(b) + [θi(X), b]g) .

By (67) and

g g g − 2LX (θi, b) + P(θi(X), b) + 2(θi,X(b) + [θi(X), b]g) (Y ) = −2(R(X,Y ), b) = −2Q(X, b)(Y ), we get

X, b ind = ∇X b − 2Q(X, b). (69) J K

36 Similarly, we have

a, Y ind = 2Q(Y, a) − ∇Y a. (70) J K For all X,Y ∈ Γ(TUi), we have

−1 T X,Y ind = Si Si(X), Si(Y ) S J K J K −1 g i g i T = Si X + θi(X) − (θi, θi(X)) − X Bi,Y + θi(Y ) − (θi, θi(Y )) − Y Bi S J K −1 = Si [X,Y ] + X(θi(Y )) − Y (θi(X)) + [θi(X), θi(Y )]g

g g −LX (θi, θi(Y )) + LY (θi, θi(X)) + P(θi(X), θi(Y )) − LX iY Bi + iY diX Bi

= [X,Y ] − θi([X,Y ]) + X(θi(Y )) − Y (θi(X)) + [θi(X), θi(Y )]g

−LX iY Bi + iY diX Bi + i[X,Y ]Bi + Ξ,

where

g g g Ξ = −LX (θi, θi(Y )) + LY (θi, θi(X)) + P(θi(X), θi(Y )) − (θi, θi([X,Y ])) g +2(θi,X(θi(Y )) − y(θi(X)) + [θi(X), θi(Y )]g) .

Obviously, we have

−θi([X,Y ]) + X(θi(Y )) − Y (θi(X)) + [θi(X), θi(Y )]g = dθi(X,Y ) + [θi(X), θi(Y )]g = Ri(X,Y ),

−LX iY Bi + iY diX Bi + i[X,Y ]Bi = −dBi(X,Y, ·).

Furthermore, we have

g g g g Ξ(Z) = 2(θi(Y ), Zθi(X)) − X(θi(Z), θi(Y )) + (θi[X,Z], θi(Y )) + d(θi, θi(X)) (y, Z) g g −(θi(Z), θi[X,Y ]) + 2(θi(Z),X(θi(Y )) − Y (θi(X)) + [θi(X), θi(Y )]g) g g = (θi(X), dθi[Y,Z]) + c.p. + 2(θi(Z), [θi(X), θi(Y )]g) 1 = (θ , dθ )g + (θ , [θ , θ ] )g (X,Y,Z) i i 3 i i i g = cs3(θi)(X,Y,Z).

Therefore, by Lemma 4.7 and the fact that −dBi + cs3(θi) can be glued to a global 3-form H, we have X,Y ind = [X,Y ] + R(X,Y ) + − dBi + cs3(θi) (X,Y, ·) = [X,Y ] + R(X,Y ) + H(X,Y, ·). (71) J K Furthermore, it is straightforward to obtain that

i X, η ind = LX η, ξ, Y ind = − Y dξ, a, η ind = 0, η, b ind = 0, ξ, η ind = 0. (72) J K J K J K J K J K

By (68)-(72), we deduce that the induced bracket −, − ind is exactly given by (23), i.e. J K T −, − ind = −, − ∇,R,H . J K J K The proof is ﬁnished.

A.4 Inner automorphisms of transitive Courant algebroids In this subsection, we prove that the automorphisms that appeared in Proposition 5.1 are inner auto- T T morphisms of the standard transitive Courant algebroid (TgU, −, − S , h−, −i , prTU ). In his letter to Weinstein [42], Sˇevera claimed that the inner automorphism groupJ KInn(U) of the standard transitive

37 Courant algebroid over U is an extension of the group of G-valued function C∞(U, G) by closed 2-forms 2 Ωcl(U), 2 ∞ Ωcl(U) → Inn(U) → C (U, G). (73) More precisely, an inner automorphism is a pair (g, ω), where g is a G-valued function and ω ∈ Ω2(U), such that dω + g∗C = 0, (74) 1 g ˆ g where C = 6 (θMC, [θMC, θMC]) , or equivalently C(ˆa, b, cˆ) = (a, [b, c]) . The group structure is given by

∗ ∗ g (g1, ω1)(g2, ω2) = (g1g2, ω1 + ω2 + (g1 θMC, adg1 g2 θMC) ), (g, ω)−1 = (g−1, −ω).

Now we give the corresponding matrix form of an inner automorphism. The matrix corresponding to (g, ω) is given by  1 0 0  ∗ Ψ =  −g θMC adg 0  . (75) ∗ ? ∗ ∗ ? ω − (g θMC) ◦ g θMC 2(g θMC) ◦ adg 1 Proposition A.10. Ψ given above is an automorphism of the standard transitive Courant algebroid. Proof. It is straightforward to see that (28)-(30) hold. For all X,Y,Z ∈ Γ(TU), we have

hLX ω(Y ) − iY dω(X) − ω([X,Y ]),Zi = dω(X,Y,Z).

sym ∗ ? ∗ Denote by β = −(g θMC) ◦ g θMC. By straightforward computations, we have sym i sym sym ∗ ∗ ∗ hLX β (Y ) − Y dβ (X) − β ([X,Y ]) + P(g θMC, g θMC),Zi = g C(X,Y,Z).

By (74), we deduce that (31) holds. Thus Ψ given above is an automorphism.

See [24, Corollary 4.2] for a similar result on inner automorphisms in another setting.

Corollary A.11. Λ01 given in Proposition 5.1 is an inner automorphism of the standard transitive Courant algebroid.

Acknowledgement

The last author thanks much Giovanni Felder for patiently going through complicated calculations together and pointing out correct directions for the most technical part of this article during her visit to ETH. The last two authors thank Anton Alekseev and Pavol Sˇevera for many inspiring conversations. We warmly thank Olivier Brahic, Zhuo Chen, Domenico Fiorenza, Fei Han, Kirill Mackenzie, Eckhard Meinrenken, Ralf Meyer, Thomas Nikolaus, Dmitry Roytenberg, Thomas Schick, Chris Schommer-Pries, Urs Schreiber, Konrad Waldorf and Marco Zambon for very helpful discussion. Y. Sheng is supported by NSFC (11471139) and NSF of Jilin Province (20170101050JC); X. Xu was partially supported by the SNSF grants P2GEP2-165118 and NCCR SwissMAP; C. Zhu is supported by the German Research Foun- dation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen, and DFG ZH 274/1-1.

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Department of Mathematics, Jilin University, Changchun 130012, China Email: [email protected]

Department of Mathematics, Massachusetts Institute of Technology, Cambridge 02139, United States Email: [email protected]

Mathematics Institute, Georg-August-University Göttingen, Göttingen 37073, Germany Email: [email protected]